Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:09  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:17  
Because we know that algorithms are super cool human achievements. But, ya'll, they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. I had to take a breath that time. Couldn't do it without. Sorry.

Kim  0:35  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:42  
We invite you to join us to make math more figure-out-able. And that also means I need to work out more or something. It's a sign. I got to get more air. Hey, Kim, I'm going to tell you today. We need to do a shout out. So, I had a meeting the other day. So, I meet with schools and districts. We work with schools and districts, and we work with our learning suite, and it's amazing what we can do. Ya'll, if you're a school district leader, you should get with us because we love working with schools and districts. And I was on a meeting the other day with Noreen. And I am going to shout out, Noreen and Cathy. Ya'll are amazing, and we are so looking forward to. We're going to start our work with them in days. And just thrilled to be working with Methuen School District. And you guys are amazing. Whoo! Alright.

Kim  1:28  
Super fun. 

Pam  1:28  
Second thing we're going to do today is I want to tell you a story, Kim.

Kim  1:34  
Mmhm. 

Pam  1:35  
So, I think I mentioned this to you the other day. I was on a plane. Which I do often. And, you know, I don't anymore, I don't talk as much on planes. I don't know if it was Covid or...

Kim  1:47  
Which is funny because every time you go on a trip, you come back and you're like, "Hey..." 

Pam  1:51  
I have a story.

Kim  1:51  
"I was talking to this person." I'm like, "I don't talk to anybody. It's my quiet time. Shh."

Pam  1:56  
Well, I don't usually. I don't know. I don't so much anymore. 

Kim  1:59  
Okay. 

Pam  1:59  
I did. I had a very delightful conversation with a businessman. 

Kim  2:03  
Okay.

Pam  2:04  
He was a sales guy. And we actually talked about some business stuff. And it was interesting. You know, because Math is Figure-Out-Able is a small business. And I'm a math teacher. I don't do business, so, you know, I'm trying to figure out how to do that. And anyway, it was super interesting. He asked me what I do, and I kind of gave him a little, you know, the spiel I usually give people. And he told me, it was terrible. I need to come up with a better elevator pitch.

Kim  2:27  
No.

Pam  2:27  
That was good. I mean, he liked our mission. He just said my elevator pitch was bad.

Kim  2:32  
Interesting.

Pam  2:32  
Which it was good input. You know, I'll take his input. But one of the things that he said was he heard me, you know, talk a little bit about how we're shifting the way we teach math, and why, and how the way mathy people think is not rote memorizing and mimicking algorithms. And he said, "Let me tell you. Let me tell you. The most important thing that was true for me as a student was that my teachers had me memorize multiplication tables." And I was like. I was kind of curious for him to say that. I was like, "Well, that's kind of against everything I believe." But I didn't say that. I'm just trying to get undercut what he's saying. And he goes, "Yeah, they gave us this table, this chart." And he kind of, you know, was moving his hands. And he's like, "There was, you know, all those facts in there." And he's like, "It was so important for me to memorize those." And I said, "I'm kind of curious what you mean by that. Say, more about memorize."

Kim  3:22  
Yeah.

Pam  3:23  
Kim, he dove into. He said, "It was, you know, the relationships between that I could realize how this one was related to that one. And these were connected to those.  And it was all this interrelated, consistent system that all was meaningful. And it had..."

Kim  3:39  
Huh.

Pam  3:40  
And I thought. And I said, "So, like, when you say memorize, and you looked at say this one right here..." And it was like, I don't know, pick one. 6 times 9 or something. I'm like, "Does that... You were like, 6 times 9 grows on the vine, and so it's..." I don't have a rhyme for this. Score, so it's 54. Like, whatever. I don't know. "Is that how you memorize it?" And he looked at me like, "No, idiot." I mean, he was polite.

Kim  4:02  
Yeah. 

Pam  4:02  
He was like, "Uh, no." And I was like, "So, when you say memorize..." And then again, he launched off into this, "Yeah, because you need to know how it's connected to like 10 times 6, and how... Or I could think about eight 6s. It's like double four 6s. It's one more 6." Kim, everything he said was about how the facts were related to each other and the connections between them. 

Kim  4:27  
Yeah.

Pam  4:28  
But he thought that he gained all that because his teacher forced him to, quote, unquote, "memorize the table". That somehow, as a student, when he was given that table of facts, it was enough for him to make those connections, and relationships, and build on them, and travel the mental path of figuring them and connecting them between each other. That's what he meant when he said, "It was so important that my teacher made me memorize the facts." He didn't mean rote memorize at all. 

Kim  5:07  
Yeah. You

Kim  5:08  
know what, I wonder if that's what he thinks quote "memorizing" is.? I wonder if like his teacher put the table in front of him, and he was like, "Oh, there's a cool puzzle thing." And like look at it and mess with it. And he was just left alone with it, and he looked for things. And maybe that's what he calls memorizing but doesn't even realize that that's not what other people necessarily do. Like, they're litterally like naming numbers over and over again. Interesting. 

Pam  5:42  
Yeah,

Pam  5:42  
and maybe skip counting to fill in the table. So, in other words, many students could hear the teacher say, "Here is this table. You need to know this." And students could do what he did. I think that's what my son did. Students could do what I did. Well, see, I was never given the table. I was given flash cards, and so for me, it was just repeated exposure, rote memorizing. 

Kim  6:05  
Yeah. 

Pam  6:07  
I did have some rhyming things for a few of them that I couldn't remember. Couldn't just remember to get the ones I didn't. But I think there are many teachers and students out there that say, "You should know this. Okay, let's see. I see these. I see the relationships of skip counting." Consider, teachers, that if you've got kids who have built some additive reasoning, and they're looking for patterns in the table, that might be the only pattern they see is the skip count. So, then they say to themselves, "Okay, I guess if I'm going to know this..." Quote, unquote, whatever, "know" means. "...I guess I'll skip count to recreate it."

Kim  6:39  
Right. 

Pam  6:39  
Then, they grow up, and they become a teacher, and they tell their kids, "You need to know this. Alright, everybody. Recreate this every time you need a multiplication fact."

Kim  6:47  
Right.

Pam  6:47  
"Recreate the table by skip counting." And we're sort of keeping kids in that additive reasoning. 

Kim  6:53  
Yeah.

Pam  6:53  
Yeah. So, it's interesting. It's fascinating to me the words "know" and "memorize" and how we connect it with "facts". And words matter. In fact, Kim, I was just on a call this morning with Sam... Oh, golly. I think her last name is Parks. She's from Number Club. And one of the things that she and I were talking about is the more that she writes... We're going to talk more about Number Club in the future. But the more that she writes about Number Club, how she's becoming... Maybe not becoming. She's aware words. That we say words that mean different things to different people, and how she's being very careful to define her terms. And I think she's about to write a blog on the word "play".

Kim  7:37  
Mmhm. 

Pam  7:38  
And, Kim, I would love to hear you. Okay, on another episode, maybe, we can talk more about like when we say "play with mathematics", I am 100% sure that what you mean by "play with mathematics" is 100% different than what I meant when I was a student in math. By "playing with mathematics". Like I would have had not even a vision of what it not even close to what you would mean. Anyway. So, words matter. And I'm just going to suggest today that "memorize". That like what do we mean when we say that? And then specifically with "facts", "memorize", does it mean to know? Oh, Kim! The other day, in one of the things that we're writing. I can't even tell you which one it is. One of the things we're writing, an editor popped back and said... I said something about "own the facts". And the editor came back with, "Why don't you just say, 'know the facts'?" Well, Kim, like do those two words mean different things to you? 

Kim  8:33  
Absolutely. Yeah. 

Pam  8:34  
Wait, say more. Like, wait. Absolutely? Can you put some words to how they...

Kim  8:38  
Yeah, I can know the answers to individual problems, but not own them in such a way that I can relate them to something else. I can't make connections. I can know individual facts. But when you own something, it's like you have... I don't even have the words. You have facility over it. You have relationships to other thing. You like know it deeply in such a way that you can...

Pam  9:07  
In and out. Uncompress. There's a word.

Kim  9:09  
Yeah, you can flexibly do things with it. Yeah.

Pam  9:12  
Yeah,

Pam  9:12  
so you might have a fact at your fingertips. But you can uncompress it and have lots of relationships and connections and deeper. Yeah, all those words.

Kim  9:21  
Yeah. 

Pam  9:21  
Alright, so if we were to say that we want kids to know their facts and more, what's one way that that could look? No, there was something else I wanted to do before I went here. One of the things that he said on the plane was something about napkin math. And I feel like, Kim, I mentioned that to you, and you were going to say something about napkin math. 

Kim  9:41  
Yeah,

Kim  9:43  
we were talking, and you said that this man often did napkin math. And I think we were talking about what does napkin math mean to you, to me? And you said, "I feel like it doesn't mean that people just grab something and like write an algorithm on it, that there's some like playfulness to it." And I said something like, "Well, even if they are writing algorithms, which I suspect maybe they're not, but even if they are, it's probably that at least they have a desire to mess." So, like, in that moment, they're like, "Oh, I thought of something. I'm going to sidestep. I'm going to do some things." Which means they probably think that it's within their capacity, that they have access to whatever it is that they like want to stop the conversation. They want to sketch something out because it's burning in them that they want to handle it.

Pam  10:33  
And they can handle it. And handling it then gives them some information they can move forward with. 

Kim  10:38  
Yeah. 

Pam  10:39  
Yeah, absolutely. 

Kim  10:40  
Yeah. 

Pam  10:40  
And I wonder how many people out there are doing quote, unquote, "napkin math" that think, "Oh, that's just a trick I taught myself because I can't really..." You know, whatever. When, in reality, they're using relationships and connections nicely. I also wonder how many people think, "Well, yeah, that's like... You know, there's school math and then there's napkin math." And I'm like, "Ah! why can't we teach that like that's real. Let's teach that!"

Kim  11:03  
Well, and I wonder how many people are at dinner or in a meeting or in a whatever, and somebody raises something, and they don't engage. Like, every single notebook I ever have in every conversation has got something sketched out on it. But I wonder how many people go, "Oh, you just said numbers to me," and they don't like... They just deflect. They avoid. They, you know, don't...

Pam  11:25  
Or they believe. They're like, "Oh, that's what you got? Okay, I have no idea how or how that's related, but I'm just going to believe." And you could be a magnitude off or you could have an extra 0 in there." 

Kim  11:33  
Sure. 

Pam  11:34  
Yeah, that was me. That was me early on. Absolutely. So, Kim, how can we help kids gain what this delightful gentleman on the plane gained?

Kim  11:44  
Yeah. 

Pam  11:44  
How can we do that? Well, one way we can do that is Problem Strings. So, I'm going to run you through just a quick Problem String that can help students not only gain the facts but gain the kind of relationships that this gentleman was talking about, and even more. Alright, so we're just going to run with what if I were to say, hey, Kim, let's picture a bag of oranges.

Kim  12:04  
Okay. 

Pam  12:05  
And that bag of oranges has 8 oranges in it. I've just drawn a ratio table, 1 bag to 8 oranges. How many oranges would be in 2 bags? 

Kim  12:12  
16. 

Pam  12:12  
And I'm going to just do some scaling arrows to just like double the number of bags, double the number of oranges. How about the number of oranges in 4 bags?

Kim  12:22  
32.

Pam  12:23  
Now, at this point, I might pause a little bit, ask kids how they're thinking about doubling 16, just to get that out because kids might not know double 16. If we're working on learning the facts, they might not know double 16, so we might just do that off to the side a little bit. How about 8 bags?

Kim  12:38  
64.

Pam  12:39  
That one I'm not going to spend time on. How about 10 bags? This might sound familiar with the Problem String we did last week. These are good relationships. Yeah, 10 bags.

Kim  12:48  
80.

Pam  12:49  
80. And then I might talk about how they got 10 bags. 8 and the 2. They might know the times 10. For sure I'm going to notice the times 10 because if kids are learning their facts, they might not know the times 10. So, once we add the 8 and the 2 bags and get 80 oranges, then I'm going to be like, "Hey, check it out. If from 1 to 10, that's times 10. Look at that 8 times 10, that's 80. It happened again." And I'm going to say it happened again because I'm going to point that out every time. Every time a times 10 happens, I'm going to point out, "Look at that, it did it again." 8 times 10 is like eight 10s. It's like 8 in the tens slot. 0 ones left over. I'm just going to just notice that times 10 thing. You're like. Okay, so I just want to point out, so far in this string, we got kids thinking about 8s. 2 of them, 4 of them, 8 of them, 10 of them. Thinking and reasoning. They're using relationships. "Pam, Pam, you want kids to do that every time? I'd rather have kids just know." I don't want them to necessarily do it every time. But I want them to be able to if they don't remember. I want them to have these relationships that they don't remember 8 times 8, they could double, double, double to get there.

Kim  13:51  
Mmhm. 

Pam  13:51  
But notice I'm building a lot of other things. So, doubling. Got the times 10 pattern happening. Then, I'm going to ask them 9. Now, these numbers are in order on purpose. I want them to be able to use what they have up there to get 9. Kim, what's one way they could use what's already up there to get 9.

Kim  14:08  
They can start with 10 packs is 80, and then subtract a pack. (unclear)

Pam  14:12  
And so, 80 subtract 8, that's 72. Cool. Could have also added the 8 and the 1. I'll probably share both of those. Then I'll say, "How about 5 packs?"

Kim  14:20  
Mmhm.

Pam  14:21  
That's on purpose.

Kim  14:21  
Which is 40. 

Pam  14:22  
That's on. Thank you. That's on purpose to come after the 10.

Kim  14:25  
Mmhm. 

Pam  14:26  
I'm definitely going to have kids have 4 packs and 1 pack. And that's okay. But I'm going to want to encourage them to think about half of the 10. You don't have to know your 5s if you know your 10s. Double. Or divide the 10 packs to 5 packs. Divide the 80... Packs. I just said packs. Haha. Bags and oranges. Divide the 80 oranges in half to get 40 oranges. Cool. So, so far, we've just, in a very quick kind of way, gotten kids to think about 2 times 8, 4 times 8, 8 times 8. 10, times 8, 9 times 8, 5 times 8. We don't have them all yet. Let's go 7 times 8. Now, here, I could have gone 6. Today, I'm going to choose to do 7 for reasons. Is there anything up there that could help you?

Kim  15:07  
Yeah, we just did 5, so we could do 5 bags and 2 bags.

Pam  15:11  
We have the 5 and the 2. And 40 and 16 oranges. Bam. That's not too bad. That's 56. Okay, cool. I might point out to kids, "Hey, do a lot of you know 5 times 8? And if you didn't, you could cut it in half. Bam. You know 2 times 8. Just look how easy it is to add 40 and 16. Sweet, there's 56." Travel that that path often. But here's where it gets fun. We don't just want to build. We want to build the single-digit facts, but not just the single-digit facts. Now, we might say, "Yum, I like oranges. How about 70 bags of oranges?" How many oranges are in 70 bags? 

Kim  15:45  
That's 560.

Pam  15:47  
And you did that how?

Kim  15:49  
7 scale up times 10. And 56 times 10.

Pam  15:53  
Nice. Then I might say, "Alright, we got 70 bags. That's a lot of oranges. How about 75 bags of oranges? I wonder. I wonder if there's anything up there that could help you. I don't know. Maybe not." Go ahead.

Kim  15:54  
The 70 bags and the 5 bags we already found make 560 and 40,

Kim  16:10  
so 600 oranges. 

Pam  16:11  
That's even a nice.

Kim  16:13  
Mmhm.

Pam  16:14  
The addition wasn't too shabby. Hey, let's go... Really like oranges. Let's go for 100 bags. 100 bags of oranges. 

Kim  16:15  
800 oranges. 

Pam  16:15  
800. I love oranges. 

Kim  16:15  
(unclear). 

Pam  16:16  
Bam. That's too many oranges, Kim. Silly. Just 97 bags. 97 bags. 

Kim  16:28  
Okay.

Kim  16:30  
Well, I want to remove 3 bags of oranges. And I don't see that you have 3 bags on there. 

Pam  16:36  
Rude. 

Kim  16:37  
I know. But I know that we have 1 bag and 2 bag there, so I can add those.

Pam  16:41  
Okay. 

Kim  16:42  
Or I can think about what's right in the middle of 2 bags and 4 bags. 

Pam  16:46  
Oh, nice. 

Kim  16:47  
So, anyway. That's 24 oranges.

Pam  16:50  
Okay. 

Kim  16:50  
So, the 800 minus 24 is 776.

Pam  16:56  
You could play a little I Have, You Need there. Kim, if you weren't as Over as you are, was there anything else that kids might have done to get the 97 with what we have?

Kim  17:08  
Mmhm. If you have 9 bags.

Pam  17:10  
Mmhm. 

Kim  17:10  
They can make an entry that's 90 bags.

Pam  17:13  
Mmm, okay.

Kim  17:14  
And add it to the 7 we already have.

Pam  17:17  
And so, like 90 bags would... If 9 bags had 72, 90 would have 720. 

Kim  17:21  
Mmhm.

Pam  17:22  
We had the 7 bags, so we could add the 720 plus the 56 oranges in 7 bags. And that could also get us 776.

Pam  17:30  
Nice. Cool. Okay, last problem. What if we were at the race, and we counted 784 oranges, and we were super curious how many bags we bought. 784 oranges. How many bags?

Kim  17:44  
Well, we just did 97 bags. And that was really close. 776 is only 1 bag away from 784.

Pam  17:52  
Mmhm. 

Kim  17:53  
So, then it must be 98

Kim  17:55  
bags. 

Pam  17:55  
98 bags. Nice. Is that also 2 bags, 16 oranges, under 800. 784, 16. Nice couple different ways to. Cool. So, Kim, we are suggesting to teachers out there that we can do Problem Strings to help kids think about 8s, that then lead into things like you just found 97 times 8 thinking and reasoning. You found 75 times 8 thinking and reasoning. And you found 784 divided by 8 thinking and reasoning, using what you know. And in the process building multiplicative relationships, building multiplicative reasoning. so, we're getting the facts, multiplicative reasoning, place value, reasonableness, magnitude. All sorts. A sense of what division means is happening. A sense of what multiplication means. A feel for operations. I wonder if a kid ever ran into a word problem that said, "I have 784 lovely oranges. If oranges come 8 oranges in a bag, how many bags do I have?" I wonder if a kid can go, "Ooh, this feels like something I've done before. I have felt this oranges and bags before. Bam. I can dive in." In other words, what they're not doing is going, "Mmm, let's see. Keyword oranges. Nope, I don't remember that as a keyword." Right? They're not doing that. 

Pam  19:15  
Yeah. They've been reasoning. They've been in a situation where their brain has traveled this reasoning path. Word problems just become... In fact, Kim, I didn't even tell you. Our editor said to me on Sunday. He said, "Hey, that last episode that I just edited..." So, it was all about word problems, which was an episode that came out just a minute ago, he's like, "I really like how you said if we get kids reasoning, then word problems we're just keep reasoning." I was like, "Yeah, that really hits."

Kim  19:45  
Right, right. I think everything that you just described that's happening in this Problem String. You know, just listen to their reasoning, and they're doubling, and they're... All of those things. I think there are kids like the man you met on the plane who probably see a multiplication table, and their brain travels through those things. But we've got to do better by giving kids experiences, so that not everybody who is doing that naturally has somebody helping facilitate that learning for them. Because they're capable. We just have to do things like these Problem Strings, where we are exposing relationships, so that they can grab on.

Pam  20:22  
Nicely said. Yeah. And even the guy who did it naturally. Why leave him to do it on his own? 

Kim  20:28  
Right.

Pam  20:28  
He could have gone further faster if we would have actually helped him. And now that we know how to do it, we can. Ya'll, I am reminded of an email that I got just the other day. I'm going to butcher his name. I'm so sorry. S. Sundaram. Sundaram. Sundaram. I'm sorry. Delightful email, who said, "Would you consider rote memorization and meaningful memorization? The difference. What is needed for fluency is the second type. Fluency is meaningful memorization, which can also be described," as he said, "remember by repeated use in the right context." I'm going to say that again. "Remember by repeated use in the right context." So, you're sort of saying memorization could be linked to we want to remember. Well, we do. We want kids to own their facts. That means they remember them. But we're going to get that memory by repeated use in the right context. And use, repeated is that what we just did in this Problem String? Repeated generating? Repeated finding? Repeated reasoning? I might say remember by repeated reasoning in the right context. But I really... Yeah, I really like that. Like, is it rote memorization? No! It's meaningful memorization if the way we're defining "memorization" is remember. It's the remembering part by repeated reasoning in the right context.

Pam  21:51  
I really like that. 

Kim  21:52  
So good. Yeah, I appreciate that redefining what "memorization" means. 

Pam  21:58  
Nice. 

Kim  21:58  
Alright, so we have another lovely download for you. If you would love to grab it, it is www.mathisfigureoutable.com/factsps. mathisfigureoutable.com/...

Pam  22:07  
You can do it. You can do it.

Kim  22:17  
I am so done. 

Pam  22:19  
I'll do it. I'll do it. mathisfigureoutable.com/factsps. Because it's Facts Problem Strings. So, it's factsps will get you Facts Problem Strings. Ya'll, 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!