Pam Harris  00:02

Hey fellow mathematicians. Welcome to the podcast where Math Figure-Out-Able. I'm Pam.

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

And we're here to suggest that mathematics class is more about mathematizing. Not about mimicking or rote memorizing, but it's about thinking and reasoning about creating and using mental relationships, that math class can be less like it has been for so many of us, and more like mathematicians working together. We answer the question: If not algorithms, then what?

Kim Montague  00:35

So for the last couple of weeks, we have been doing an important series about multiplication facts. So just to summarize, in Episode 34, we shared some clarification around the words memorization, automaticity, and fluency. Kind of big ideas. In Episode 35, we described some things that teachers often do in the name of helping kids learn multiplication facts that actually do not work like we want them to, because they're focused on either learning the facts in isolation, using shame to motivate or emphasizing speed. And then in Episode 36, we shared one of our favorite ways to build facility with facts using Problem Strings. So in - 

Pam Harris  01:16

Favorite favorite way, yep. 

Kim Montague  01:17

Yeah. So in today's wrap up, we want to share some more ideas about what we would recommend, as you're working with students and their multiplication facts. So this one is super fantastic for both parents and teachers.

Pam Harris  01:30

Yeah, in last week's episode, number 36, you said, Kim, that you felt the most important thing you did was putting multiplication at the beginning of the year to give it lots of time. 

Kim Montague  01:41

Yeah. 

Pam Harris  01:41

Because then you could do it all year long. But I think one of the most important things that you did in your teaching was your attitude about multiplication facts. You were also in it for the long haul for the whole year. And you were committed to giving kids lots and lots of opportunities to experience facts. That attitude showed because you didn't freak out if a student didn't just know a fact. The expectation that you had was that if they didn't just know it, they were supposed to figure it out. I think it's so important. 

Kim Montague  02:14

Yeah, yeah. I mean, I could have provided students with an answer, right? If they didn't know one, or have them look on a chart, maybe on their desk, or on the wall.

Pam Harris  02:22

Like you could have asked it and be like, alright you don't know, how about somebody else? Like, you know, just skip over. Nope.

Kim Montague  02:27

But listen, I don't think we emphasize this enough. When kids are figuring out something, that's when the learning is happening, right? It's not about the answer. It's about the mathematizing they're doing, and in this case, building their Multiplicative Reasoning by figuring out the facts that they don't own.

Pam Harris  02:45

Yeah, absolutely. They get the facts. They own the facts. 

Kim Montague  02:49

Yeah. 

Pam Harris  02:49

But as a result of the thinking and the reasoning that they've been doing, and then we also get, correspondingly, we get them reasoning multiplicatively. 

Kim Montague  02:59

Yeah.

Pam Harris  03:00

It's so important that we are building their reasoning. Y'all, if you want more about why it's so important to build their Multiplicative Reasoning, or all of their reasonings. Check out episodes five and six, where we really get at Developing Mathematical Reasoning and why that's so important because it is more important that we have multiplicative reasoners, then we have the facts. But if we develop multiplicative reasoners, then they also have the facts. So we get both in the same bang. So interesting, when I walk around the country, and I ask middle school students everywhere, a most missed fact, like I just grabbed the most missed fact, seven times eight, six times eight, nine times seven, like pick a most missed fact.

Kim Montague  03:39

Yep.

Pam Harris  03:39

I primarily get one of two answers. Either that kid will look at me, they'll look me in the eye. And they'd be like, "Huh, I don't know." Which means they think the fact is know or not knowable. That's huge, right? They don't think the fact is Figure-Out-Able and math is Figure-Out-Able. Facts are Figure-Out-Able. So we want to make sure students know  it is your job right now to figure it out. I didn't ask you if you knew it, I asked you what it is. What do you know and figure it out from there. Or the other thing that kids will do is they'll skip count because the teachers have left them too inefficient. But we don't want them skip counting. That's Additive Reasoning, unless they're beginning right? If you're like at the beginning or wherever kids are, if that's what they're doing, don't slap their hand. It's not about shaming. It's not about saying, "Don't do that." It's about like, helping them build to do something more productive, more sophisticated. Alright, so I just said that one of the most important things I think you did was your attitude. Let me tell you how I know that, A: you had my own personal kids. But, B: you know, you were their teacher. Thank you. Thank you. Thank you.

Kim Montague  04:41

Not stressful at all. 

Pam Harris  04:45

Come on!

Kim Montague  04:46

You were great.

Pam Harris  04:49

Anyway, so I was working with teachers in the district and one day, some teachers told me Hey, Pam, we're a little concerned that teachers are miss-hearing you and they're hearing you say don't have the kids rote memorize their facts. They're hearing that they don't have the kids learn the facts at all. And I was like, Oh, no, not what I mean, I absolutely want the facts at their fingertips. They just want to get there through relationships. And so, in that concern, I decided to go video kids to see you know how the facts were going. I wanted to prove some evidence. So I grabbed some of your students, and it was fascinating to see what they did. Y'all Let me tell you the first most important thing I want you to remember, every single kid that I asked in her class, what is - I asked them six times seven. I asked them 11 times 12, those are for sure two, and then maybe more - every single one of those kids either told me the answer, or figured it out, and then told me the answer. What they didn't do, none of them, not even the most struggling kid in that room. None of them said, "I don't know." Like not a one. They knew their job was to figure it out. None of them were panicked. None of them were nervous. None of them freaked out. None of them like, oh, like I should know that, I'm sad. None of them recited a rhyme or sang a song. And nobody, like had those crazy posters we talked about last time. Like, every single one of them knew it was their job to tell me what the product was. And if they didn't just know it, they figured it out in a multiplicative, mathy kind of way.

Kim Montague  06:16

I'm so proud. I didn't know that.

Pam Harris  06:21

That's news to you, huh? No, it was fabulous. And that's what we're after. Right? We are after that kind of automaticity with kids or that they know, their job is to figure it out. And every single one of them were able to get to the correct answer. Some took longer. It was 4th grade. They're still learning the facts.  So really important is for everybody to realize that that attitude towards facts - in fact, let me just say parents, so often when Kim and I are like talking with your kids, and you're standing there, one of the things that we - we won't do it, of course - but we are tempted to like put our hands over your mouth just a little bit. 

Kim Montague  06:58

Yeah. 

Pam Harris  06:58

Because we'll ask your kid a question. And the kid starts to think because they're like clear, because our attitude toward it is: figure it out. And the parents are like, "You know that, you should know that. Come on, you know that." 

Kim Montague  07:06

Yeah.

Pam Harris  07:07

It's like your self esteem is on the line. And so you're like, "Come on, kid, like show that we're good in this family, we can do the math thing." I would much rather have you let your student think. Give them the time, like don't panic, don't panic for them. That causes their panic.

Kim Montague  07:23

Yeah, the thinking is so important. So you asked what were some of the things that I focused on. And we have already mentioned the order of the sequence for the year.

Pam Harris  07:32

So last episode, we talked about that, uh-huh.

Kim Montague  07:34

And the order of sequence for the facts that we introduced. And so one of the routines that we did to work on those relationships were basic fact Problem Strings, and if you know anything about Pam at all, you know, she's a huge advocate for Problem Strings. And last week's episode was all about using Problem Strings to build single-digit facts. Also, you may have seen a facebook live where she's done a string. And if you'd like to see what that might look like, you can check out the playlist of Facebook Lives in her Math Is Figure-Out-Able YouTube channel, or in the show notes, where we will put some links.

Pam Harris  08:07

Yep, absolutely. Besides Problem Strings. Kim, I know that you played a lot of games throughout the year. 

Kim Montague  08:14

Yeah. 

Pam Harris  08:14

So I'm gonna talk about a couple games that I saw you play with your students. 

Kim Montague  08:17

Okay. 

Pam Harris  08:17

One of them was a game from Investigations of Data, Number, and Space. And they were all about arrays, like the area model. And so we really liked the fact that in that curriculum or textbook series, they had these rectangles that represented the facts. So if it was a four by nine, then it literally looked like a squatty rectangle that was four rows by nine columns. And then there were several games that you could play with those. And they had all the facts that were represented on these arrays. So if you looked at this array card deck, it was these several different shapes and sizes, because like seven times eight, was bigger than six times eight, was greater than five times eight, right? But we had all of those rectangles in those dimensions, with all the same sort of unit length for all of them. And then we could play lots of games with. One of the things we liked about those games is that they were very differentiateable. So one of the games that I talked about just a little bit, because it's so naturally differentiated. And that's one of the things we like about games, is that you just sort of put the cards out on the floor, and just kind of like in no order, and they're just kind of all over the place. And then a kid looks for one that they can figure out and so maybe I'm a kid, and I see a four by eight, and I pick up that four by eight, and then I say, "Ooh, the product is 32." And then I put that card in my pile or whatever. And then my partner might go and my partner might look at some of the cards that are flipped over where I don't see the closed array, I don't see all the grid marks. So when I saw the four by eight, I saw the grid marks that showed four rows by eight columns. But the other kid might see a card that's flipped over that it says 36 and then that kid just sees an open array and just the number 36 is their job is to think which factor pair am I looking at? Well, that's really cool. Because if it's a square, then they should guess that that's six by six. But if it's a rectangle, then they should guess that it's a four by nine or a nine by four, or maybe even a three times 12. Or if the twelves are in there. Like they have to sort of think about, if it's a square, okay, it's got to be 6 times 6. If it's a rectangle, well, how squatty is that rectangle, how squished is it? Like a four by nine? Is it a three by 12? That was brilliant. And so see how naturally differentiated that was because kids could choose kind of a closed array look, where they're looking at like kind of what's the fact. Or the open array look where they're looking from the product to what could the factor pairs be? I loved it, that was a great game. So another one that I saw you play that we have been since done lots with is called the Product Game. And it has different names. But this is what it looks like. It looks like a tic tac toe board, where it has lots of products, each square is a product. And then they can choose from factors to create those products. And then they choose the factors, then they put an X in that product, and they're trying to get four or five in a row. But there's a lot of strategy involved, because when you're choosing the factors to then get your product, you can only - the game starts out with two markers. And you can only move one of the markers. And so if the first player puts them on five and seven, and then puts their X on 35, then now it's my turn, I can only move one of the markers. So my next product is going to either have to have a five in it, or it's going to have to have a seven in it. And so I start looking around the board at different products to decide, well, you know, where am I going to move my marker to choose which number because I might want to block you. And oh my gosh, the levels of strategy that get involved. Sometimes kids are thinking two or three moves down the road, which is again brilliantly naturally differentiated. If I'm a kid that doesn't know any facts, I'm kind of randomly choosing two products, I go see if I can find their factor. And then if it's already taken, okay, I choose another one. And if it's not taken, then I put my mark there. But if I'm a student who knows the facts, then I can play with that student because I'm thinking three or four moves ahead. 

Kim Montague  12:11

Yeah. 

Pam Harris  12:11

Well, if I move my marker there, then that will allow them to move the marker there. And then they can get that product and they can't block me and all the things. So yeah, we really like that one. We will put a link to these games in the show notes. So if you want to check that out, check out the link in the show notes to know more about those. So Kim, these are great games, will you tell us how you choose a game because I've actually heard you say - if you guys haven't picked up on it by now Kim's a little snarky. And I have actually heard you say that you think that there's a lot of games out there for kids to learn their facts and about 50% of them are any good.

Kim Montague  12:46

I might have said that. And actually, I'll definitely share some things. But before I do that, I want to mention two other games that I did not play in my classroom, because they, to my knowledge, were not available until just like the last year or two. 

Pam Harris  12:59

Gotcha. 

Kim Montague  13:00

Prime Climb is a huge, huge game that I absolutely love.

Pam Harris  13:05

You say huge, you just, you love it hugely. 

Kim Montague  13:07

I love it hugely, it's not a big game.  Big love for Prime Climb. And I'm not going to really describe it because you can search it and get a great description but Prime Climb. And actually -

Pam Harris  13:19

That's a for purchase game. 

Kim Montague  13:20

It is a for purchase game if you have a classroom wish list, if that's ever a thing for you, I don't know, but Prime Climb might be a game to purchase. But more recently, this Christmas, I got my younger son, the game Multi. And it is phenomenal. I absolutely love this game. I've mentioned it several times to Pam before. And it is like a double layered Tic Tac Toe board. And kids are thinking about multiplication the entire time. It is so fantastic. So one of the reasons that I love these games is because they have a lot of the things that I consider fantastic for games. So let me just mention a few things. One, everyone can enter into the game. Whether you're additively thinking and you're skip counting, you can still play. That's a huge component for me. I do not like games, where you know, only some of your kids can play but the other ones have to go play something different because they're not quite ready for it. 

Pam Harris  14:21

Everybody can tell, right? It's real visible who's on what game.

Kim Montague  14:24

Yes, that's so shameful. I also love games that are naturally differentiated. I love simple instructions, right? I do not want to spend an entire class period learning to play a game and modeling it and then we never get to play again. So you know time is important and we don't have a ton of it. So I want simple instructions.

Pam Harris  14:50

And simple instructions is helpful too because then you can send kids off to play it and you're not worried that they are like, "What were the rules again?" And they're looking around, you know and they get frustrated. 

Kim Montague  15:00

Yep. And I think one of the other things that I think about a lot is that a game of strategy is almost required for me. I do not want kids playing a game where it is just you go, I go, you go, I go, and they don't have to do a lot of thinking. So games that they're working with facts in the midst of doing something bigger, something strategy is really a big choice for me.

Pam Harris  15:23

And fun. That's what makes it fun. Yeah, absolutely. Yeah. And again, also differentiated, because you can have kids that are just sort of working on the thing, but you have kids that are totally using strategy.

Kim Montague  15:33

So I think we mentioned this before, but I am absolutely not a fan of games that have time elements. Like there's some sort of speed component to it. We obviously feel like kids need time to think. And the more they're exposed, and the more experience they have with their facts, the better they're going to be. If I can, I'm going to mention that there is a computer program that I think had some really, really good things about it. But there's a portion of it that one of my sons absolutely avoids. And it's one of those, it doesn't matter what the actual game is. But there's a lot of programs that use this. But it's one of those things where a fact will move across the screen, and you only have a certain amount of time to enter the answer. And just the act of adding a time component to it sends him into a tizzy. And he actually knows his facts. Right?

Pam Harris  16:31

He is a great kid. Yeah. So I remember you saying, "There's this bit in this app that he just keeps skipping." And you're like, "I'm gonna I'm gonna poke in on that. I'd like to see see what's going on." You came back, you're like, "Oh, of course." Like, it's stupid. It's pretending that being good at the facts is all about being fast with the facts. And that's not what we believe, or what we want to promote. And interesting. So interesting that you're awesome kid who mathematizes with the best of them was like, "I'm not doing that."

Kim Montague  17:00

Yeah. No stress, right. We don't want to do things that are stressful.

Pam Harris  17:04

Yeah. So remember, we want to take the long view, listeners. We want to have this idea that it's not about memorizing facts, like one shot or doing one kind of a task. We've played the game once you should know your facts. We want the long view, we're going to develop the relationships and develop the facts over time. All year long play games and do the routines that we talked about last time to help build Multiplicative Reasoning and the facts at the same time.

Kim Montague  17:31

Yeah. And so once we had some time to develop some of those ideas, when we reached the part of the year where I felt like I'd given facts their due. And I knew we could sort out ones that kids knew at their core and ones they didn't own yet.

Pam Harris  17:45

Hey, Kim, I gotta be honest with you. I'm looking at the time, this is already longer than we intended this episode to be here. So I know we said at the beginning that this is the last episode of this series. But I'm going to make an executive decision as we're recording right now that we're gonna do one more next week for the second half of what we plan to talk about today. 

Kim Montague  18:04

Do it.

Pam Harris  18:05

So next week, stay tuned for the last episode. Or maybe I should just say the next episode in the series, because I don't know how long this is going to go. So let's finish today out with remember everybody that it's about taking the long view, play these games, but play them a lot, play them over and over and over again. This is a process. 

Kim Montague  18:25

Yeah.

Pam Harris  18:25

And your attitude makes such a big difference. Use games and routines all year long. Next year, like keep going, do have your kids practice, but without the shame, without time pressure and not just learning facts in isolation.

Kim Montague  18:41

So good. We would love for you to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore problems with the world.

Pam Harris  18:49

And speaking of the world, let's change the way across the world that we teach mathematics. So will you please share the podcast with your friends and colleagues, with your parent friends? And also if you don't mind giving us a rating and a review on your favorite place where you listen to the podcast because that helps more people find the podcast. So y'all if you're interested to learn more mathematics and you want to help yourself and your students develop as mathematicians, then the Math is Figure-Out-Able Podcast is for you. Because Math is Figure-Out-Able!