Pam Harris  00:02

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

Kim Montague  00:10

And I'm Kim Montague.

Pam Harris  00:11

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 mathematical relationships. That mathematics class could 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? Alright y'all for the past four weeks, we've been doing a really cool series about multiplication facts. In Episode 34 we talked about memorization, automaticity, fluency, and helped clarify kind of the whole issue. In Episode 35 we talked about some things that teachers do sometimes that they think will help students learn the facts that actually don't work so well, because they're based on learning the facts in isolation, using shame to motivate or they emphasize speed, which is not really mathematics. In Episode 36, we shared one of our favorite ways to help students build facility with facts: Problem Strings. We love Problem Strings. In last week's Episode 37 we talked about how important your attitude is about learning the facts, and some great games to play with kids to help them build their facts and multiplicative reasoning at the same time.

Kim Montague  01:29

Yeah, we had so many great ideas we decided to keep going this episode. We wanted to share some more ideas about what we really recommend, as you're working with students and their multiplication facts.

Pam Harris  01:41

Yeah. So in the last episode, we talked about taking the long view that it's not this short, it's not one time, it's not just like, do this one thing. But it's like, let's work on these for a long time. Let's keep talking about it. In fact, Kim, you have a seventh grader? 

Kim Montague  01:56

Yep.

Pam Harris  01:57

And you still mess around with facts with him, right? 

Kim Montague  01:59

Sure

Pam Harris  01:59

Like it's a constant conversation. Now when I say that somebody might be like, "Oh, because he doesn't know them yet?" No, no, I mean, the kid is a brilliant mathematician. You guys should see the way this kid mathematizes. But it's still like a thing. It's a thing to continue to play with. Like we just keep with it. We take the long view.

Kim Montague  02:16

Yeah, to see how many different relationships kids can use, right? 

Pam Harris  02:19

Absolutely.

Kim Montague  02:20

So at some point, though, we need to dial in a little bit. So we play games and do activities to work on exposing kids to facts. 

Pam Harris  02:28

Sometime during the year, we need to dial in. 

Kim Montague  02:30

Right, right. So at some point, right, when we reach the part of the year where we feel like we're given facts their due, I would have kids think about sorting out ones they knew at their core, and ones that they didn't own yet. So we've done a lot of work, we've done a lot of activities. And then when we reach that part of the year, we make individual sets of clue cards based on the relationships that they have made use of. And those clue cards become their own personal flashcards to work with, and are based on what each student chooses.

Pam Harris  03:03

So Kim, I think I've heard you say that you actually interviewed each kid, and you would sit down with them and you sort of use like a typical flashcard deck and just say, "Hey, let's just check these out." Now remember, everybody, remember listeners, she's doing this later in the year after they've done lots of work already building the facts. And then she's like, "Hey, which of these do you just know? Do you own? You've got down deep? And which ones do you not know, yet?". Growth mindset language in there. We don't know -  'yet'. But we're gonna, and you would just sort of like show them the card and you make a pile of facts. If Kim's a student, then we have facts Kim knows, and facts Kim doesn't know - yet. And then take those facts that the student doesn't know yet and make that personalized set of what you call clue cards. 

Kim Montague  03:48

Mhmm

Pam Harris  03:48

So what does that look like? Well, let's take like a most missed fact. So say the particular kid that we're working with, doesn't know 8 x 7, or 7 x8. So you would make a personal card for that student just on an index card or whatever kind of card you have handy. On that index card, you would write '7 x 8', '8 x 7', and then a clue to help that student figure that fact. A relationship that would help that student figure that fact. So if it is 8 x7, that clue might be something like, if I'm trying to get eight sevens, then maybe I would write a clue of '7 x 7'. Because if I have seven sevens, I can use that to help me find eight sevens. I just need one more seven. And so on that card, where I've got 8 x 7, 7 x 8. Then underneath that, I might write '7 x 7'. I might write '7 x 7 = 49'. I might write '49'. Like whatever it is that kind of pings for that kid to go, "Oh, yeah, that's right. I could use that to help me think about the fact I'm trying to find" or maybe another clue might be to get eight sevens. I might write '4 x 7' or just 'four sevens', or '4 x 7 = 28', I try to write as little as possible. 

Kim Montague  05:04

Right. 

Pam Harris  05:05

So that it is just enough to ping the kid to be able to think about, "Oh I can double four sevens to get eight sevens." Or for 8 x 7, 7 x 8, a kid might say, "Well, I know my fives, like 5 x 8, I can do that." So to find seven eighths, I've got to find five eights, and then add two eights. So we might just write down like '5 x 8', or we might write '(5 x 8) + (2 x 8)'. Or we might just write '40'. And then say like, whatever it is, again, that would sort of cue that kid. Maybe to find eight sevens, we might write down '10 x 7'. Because to find eight sevens, maybe if I know 10, sevens, that's easy, right? That's 70, then can I get rid of two extra sevens? Let's just get rid of 14, oh yeah, that's 56. I might use the clue of seven sevens to find eight sevens or eight eights, to find seven eights. Because kids know those square numbers. A lot of kids know 7 x 7 and 8 x 8. And they don't know the one in between, so funny. So whatever it is that works well, for that particular student, I'm going to write on the bottom of that card. 

Kim Montague  06:11

And they've been thinking about these relationships already, right? It's not like, "Oh, let me just come up with something random." They've been working with these relationships in all the games and activities that we've been playing. 

Pam Harris  06:24

Absolutely. And so we're just looking for one that works particularly well for the student. so that when they are then practicing, because then you're going to have kids practice with these cards. 

Kim Montague  06:32

Yeah, absolutely. 

Pam Harris  06:33

Yeah. And let's talk about that for just a second. When you're having kids practice with those cards, I've heard you talk about circulating during that practice time. Can you say more about that?

Kim Montague  06:41

Yeah, so I'm listening for kids who are not making use of the clues that they said are useful to them, right? When they make these cards, then they are choosing the clue, you just gave us a whole bunch of relationships that could be used, but they come up with one that's useful to them. And then when I hear them not make use of them, they get stuck on one for quite some time, then we re-evaluate the clue that they have on their card.

Pam Harris  07:09

And you might choose a different clue. And you might  scratch that one out and put a different thing on there that maybe they could use more easily. Y'all, if you don't learn anything else, teachers, notice how important it was for Kim to be involved. Like she's circulating, and she's listening in, it's not just like, "Okay, go off on your own. And I'm gonna grade papers over here." You're constantly interested in where students are on the Landscape of Learning, you're constantly interested in how you can help nudge them towards more specific, sophisticated thinking. And I love that, so I wanted to point that out. Okay, so I'm picturing this deck of cards I'm making for each individual kid. And so I've written '7 x 8, 8 x 7'. And when I say that, y'all, I mean both, so I write '7 x 8', and then underneath that, I write '8 x 7'. And then underneath that a little bit with some space in between, then we'll write the clue that the student is going to use. And then flip it over and put '56' on the back?

Kim Montague  08:01

Oh, actually, no, no, no. So the clue goes on the front. But nowhere on the card does it have the answer. 

Pam Harris  08:08

What?

Kim Montague  08:08

I know it's totally different than a regular flashcard. But here's the deal. We're encouraging kids to play with a partner, right? They're not playing alone. So I paired them up with kids that they could have a conversation with, right? And they're not able to treat it like a regular flashcard, they had to focus on the clue, right? So if they're playing alone, and they're just like, looking at the front, and they could flip it over and look at the back, then it feels a little bit like, "I'm just asking you to memorize," which is not the goal. 

Pam Harris  08:37

Because they could just literally go: seven times eight, flip it or whatever, 7 x 8 is 56 and kind of do a memory thing. Now I'm trying to rote memorize this. And that's not what you're trying to encourage. 

Kim Montague  08:46

Right. I'm trying to encourage thinking about a relationship that makes sense to them. And with more time seeing relationships, then it will become automatic.

Pam Harris  08:54

Yeah, absolutely. Nice. So you don't actually put anything down? The answer doesn't show up there. So how do you make sure that you don't have kids that are practicing the wrong answer, you know, maybe they're talking about it, they both agree that 7 x 8 is 55 and they move on, like, how are you - ?

Kim Montague  09:11

Well, I'm gonna visit with them, right? So at some point, I'm going to sit down with kids pretty often. I've mentioned before the two thirds/one third way that I split up my class, and in that 1/3 time, you know, we know our kids so well, right, we have conversations with -

Pam Harris  09:25

Well, you do because you have conversations with them, and you're circulating and you're dropping in and you're pulling them like you just said so - 

Kim Montague  09:32

Yeah.

Pam Harris  09:33

Yeah, so keep listening, guys.

Kim Montague  09:34

I'm going to meet with my students who I know are struggling with some facts. And I'm going to have a conversation with them. And I'm going to go through their clue cards with them. And I'm going to be listening for which ones of the clues are helpful and which ones they're coming up with an incorrect product.

Pam Harris  09:49

Yeah, so you're involved. 

Kim Montague  09:50

Yeah. 

Pam Harris  09:51

You're helping that by being involved. Cool. Very nice. Alright. So let's give the listener some practice at this. So Kim, I'm going to give you a most missed fact. 

Kim Montague  10:00

Okay. 

Pam Harris  10:00

And I'm going to ask you to give a few clues that might be good ones that a teacher might offer up to a student. So when a kid sits down, and they have this most missed fact and then he's like, "I don't know." She says, "I don't know what clue to use." You might say, "Well, do you know, blank? Or do you know -?" So teachers need to have these clues, these relationships at their fingertips. What were you going to say?

Kim Montague  10:21

Well, I was gonna say that - so you said offer up. And there are definitely times where I'm making a suggestion, but I don't think one of the things that we've mentioned is that a lot of these suggestions come from the students in the class. So if I know I'm working with a most missed fact, that's going to be on many kids' lists, then we're gonna make a poster together. And it might be, you know, 8 x 7, like you said, and I'm going to ask them, "What are some clues that could go on your card?"

Pam Harris  10:49

Let's brainstorm, whole class? 

Kim Montague  10:50

Yes. And so we're brainstorming the whole class, and we're coming up with a whole chart. And then when I'm sitting down with students, I might say, "Hey, was there one on that chart that feels right to you? Is there one that you want to gravitate towards?" And so I might guide a little bit, but a lot of the times it's coming from the students; the class as a whole.

Pam Harris  11:11

But teachers - don't you feel like Kim, it would be important for teachers and parents to have a few relationships -

Kim Montague  11:16

Oh for sure. 

Pam Harris  11:16

- in their heads so that they as students are talking about it, they can hang with it. They can draw them out when they're making that class poster. They might be like, "Well, did anybody think about, you know, a little over relationship?" Okay, so let's brainstorm what some of those relationships could be for a most missed fact, like, I'm going to give you nine times seven.

Kim Montague  11:35

Okay, 9 x 7. So a really common clue that I would have seen for 9 x 7 is something really close to 10 x 7, a lot of kids just know 10 x 7, because they have seven 10s. And so I would have seen a couple of different variations. So the clue might have just said '10 x 7'. Or it could have said, if they need a little more nudge '(10 x 7) -  (1 x 7)'. A lot of kids are really comfortable with fives. So for 9 x 7, it might have said '(5 x 7) + (4 x 7)', or maybe just '5 x 7', that might have been enough. You mentioned earlier that a lot of kids love the fact that they know squares. And so you might have seen 7 x 7, or (7 x 7) + (2 x 7), could have been (9 x 9) - (2 x 9) - 

Pam Harris  12:31

Cool. 

Kim Montague  12:32

- to get to seven nines. There's a whole bunch of them, right? 

Pam Harris  12:34

Would you ever think about 3 x 7 and triple it?  Ah-huh. Or is that - I mean that's  a great one, except I'm wondering if - 

Kim Montague  12:42

Tripling 27 is kinda a funky -

Pam Harris  12:45

Ah, but it'd be 21 right? Three times seven is 21. And then triple is 21.

Kim Montague  12:50

Oh yeah, in my head it was 3 x 9. So yeah, yeah, yeah.

Pam Harris  12:55

So interesting, if you don't know 9 x 7, I'm not sure 3 x 7 is 21, tripling that? I don't know if a kid would have access to that, but it's fun to play with. And if a kid says no, that would not help me, well, then don't put that on their clue card. Right?

Kim Montague  13:09

And you know what, I want to mention that one time we were in a workshop, and we were talking about clue cards, and it's so interesting to me, because we made one clue poster together. And then I asked participants to make at their tables to make a poster of suggested clues. And then we walked around the room and I asked the participants to put a little asterisk or a star next to the clue that they would choose for that one.

Pam Harris  13:36

Let me be clear, each group had a different most missed fact. Is that right? 

Kim Montague  13:39

Correct. So we hung the posters on the wall, and one poster might have said six times eight, and one might have said nine times six. And so we had all these posters. And then we literally took a marker and everyone walked around the room. And I just said, "Hey, choose the clue that you would put if you were making a clue card for that fact." And there was overwhelmingly on every poster, almost every clue was chosen.

Pam Harris  14:04

Oh, interesting.

Kim Montague  14:05

Yeah. You would think one clue was picked more than others. 

Pam Harris  14:09

Was more popular. 

Kim Montague  14:11

Absolutely not true.

Pam Harris  14:13

Huh? That's really cool. Nice. Okay, so you ask me one. Give me a most missed fact.

Kim Montague  14:18

Okay, you take 8 times six,

Pam Harris  14:22

Okay, 8 x 6. So I might think that kids might use the double, double, double that we talked about a couple episodes ago, where they might find 2 x 6 to get 4 x 6 to get 8 x 6. Or they might just know 4 x 6. And that's delightful because 24 then is so easy to double to get 8 x 6. They also might use squares like you mentioned earlier, so like 6 x 6, and then they have to add two more sixes. I don't know that they use 8 x 8 and get rid of  - that they have eight eights, but they only need six eights, they would get rid of two eights. But we could see if they want to do that. And then again, they might use their fives. Fives are nice. They might think about five eights to get six eights. 

Kim Montague  15:06

Yeah.

Pam Harris  15:06

Yeah. Cool. So Kim, one of the things that teachers also ask us when we talk about clue cards is - we just got this the other day. What if - you know, we don't want shame, right? We've talked about how shame is not a thing that we desire, it doesn't motivate kids well. Teachers have asked, so what if kids have these huge card decks? Like, we're just really worried that some kids will have these gigantic decks, some kids have these really tiny decks. Like, how do we keep the shame out of that? 

Kim Montague  15:33

Yeah, we do get this question a lot, right? And so one of the things that I'm maybe not as clear about is that this idea of making clue cards comes really late, right? And so there may be kids who -

Pam Harris  15:45

Late in the year. 

Kim Montague  15:46

Late in the year, maybe like, you know, second semester, after we've played a lot of games and done a lot. And so there may be kids that have none or just a few, right? Or there might be some kids that have a handful. But I never, you know, for all those years never had kids who had 25 cards, or 40 cards. And so it wasn't really ever a thing, where there was this whole unbalanced situation. I also loved sending those sets of cards home. So that was a kind of thing to let parents know, these are the facts that your kids could use some more work on. And once we set up the routine at school, then that was a great way for parents to get involved or for kids to work with a sibling. I know that's not always the case for you know, for some kids. And then I knew those families well, and I would play with the kids.

Pam Harris  16:43

Nice, nice. So kids have been tinkering all through semester, tinkering and tinkering. And so by the time you got to making decks of cards, I heard you actually say that you never had kids that have more than 10. Like you know, 10 was sort of the max that a kid might have. And if you think about it, you guys, once you have like, two to 10 cards, you kind of can't tell the difference. I mean, the stacks are pretty much you know, there's so few. And there really aren't that many more to work on, because they've already been tinkering all year long, very cool.

Kim Montague  17:16

And I would suggest that maybe if you've got kids, who've got 30 or 40 cards out of the whole, you know, set, then it might be that there's some things that can be done prior to doing clue cards. Maybe there's some more routines or activities that you could do to give them more facility with those facts.

Pam Harris  17:33

Yeah, nice. Nice. Cool. So clue cards are a great thing that you guys could use to help your students after you've been tinkering a lot with relationships to help them automatize those facts. A couple other things we wanted to mention today. So our friend Berkeley Everett has these really cool things called Math Flips. Math Flips, and they are definitely worth checking out. So I just want to shout out to those today. And we'll put a link in the show notes. Those are some of the things that you could be using to tinker around with the relationships. One more that we want to mention today is another game actually, that we've just gotten wind of and I've been playing it. Kim's kids have been playing it, we've had other people playing it and we love it. I am always very hesitant to recommend, especially computer games, because so often they just go awry, and they have that time element in them. Anyway, we just don't recommend them very often. But we like this one. It's called Stick and Split. Yeah, so stick and split, I highly recommend you guys check it out. We know that right now you can get it in the App Store. It's very interesting. I'll just take a second to just explain part of what we like about it. It comes at learning multiplication, and sort of these multiplication tables or modification facts, not from one number at a time. Like I'm gonna learn my sevens today, or I'm gonna learn my fours today. Or I'm gonna work on my fives today. But it comes at it from the product side. Very much like the product game that we talked about in last week's episode. It gives you the product and you mess around with all the ways to make 15, or all the ways to make 21. And you learn sort of the factor pairs and how they relate to that product. And so we really like it. Here's a moment in time that told us we had found a good game. When Sue's son walked in the room, who was very -what's the word I want? Snarky? I don't know. He's picky. Picky is a good word. He's picky when it comes to games. He will not play a game if it's just obviously I'm supposed to learn something from this game. He wants to play games that're fun. He will play stick and split. And one day he walked up to Sue not after having played for very long and he goes, "Hey Mom, Mom! I learned a new way to make 15!". Oh, love that. I mean this is a kid who's not even in multiplication yet. He's a second grader and is already playing with ways to make 15 multiplicatively. It's fabulous. So highly recommend that you check out Stick and Split, I think we're going to expect to see really cool things from this game maker in the future. So check that out. Remember y'all It's about taking the long view. This is a process. Your attitude makes such a big difference. Use games and routines and talk with kids and use clue cards all year long. We do want kids to have practice but without shame, without time pressure, or not just teaching facts in isolation. 

Kim Montague  20:24

We would love it if you would join us on MathStratChat on Wednesday evenings. You can find that on Facebook, Twitter and Instagram, and we explore problems with the world.

Pam Harris  20:33

Yeah, and if you wouldn't mind, please share with your friends, your colleagues. We want to spread the good word that Math is Figure-Out-Able. Give us a rating and a review on your favorite podcast place so that more people can find the podcast. Y'all if you're interested to learn more mathematics and you want to help yourself and your students develop as mathematicians, then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!