Pam Harris  00:03

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

Kim Montague  00:09

And I'm Kim.

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 math 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? 

Kim Montague  00:37

Yeah, so for the last two weeks, we have been talking about an important series about multiplication facts. 

Pam Harris  00:44

Da-tah-dah.

Kim Montague  00:44

We shared some clarification around the words memorization, automaticity, and fluency. And then, last week in Episode 35, we described some things that teachers often do in the name of helping kids learn the multiplication facts that actually don't work. Like we want them to, because they're focused on either learning the facts in isolation, or using shame to motivate or really emphasizing speed. 

Pam Harris  01:10

Yep.

Kim Montague  01:10

In today's episode, we want to unpack some ideas about what we would recommend, as you're working with students and their facts. This episode is definitely for parents and teachers alike. 

Pam Harris  01:22

Because remember that the issue here isn't whether students should own their facts, we absolutely want students to own the multiplication facts. But how this owning is achieved: by rote drill and practice, or by focusing on relationships? 

Kim Montague  01:37

Yeah. 

Pam Harris  01:38

Remember, we quoted elementary mathematics researcher, Cathy Fosnot, as she talks about helping students learn the facts, by developing relationships, because in that way, the facts become automatic, but the relationships, the heart of mathematics are not sacrificed. And we totally agree with that. We have a super cool, helpful download for you for this episode. So check out the show notes or keep listening, and we'll give you the link at the end of this episode. 

Kim Montague  02:03

Yeah. So Pam, I like it when you talk about taking the long view. And I wanted to mention that one of the most important things that I think I did as a third grade teacher, in regard to facts, was to advocate for starting the year with multiplication, rather than the pretty traditional few weeks of place value, then addition and subtraction. And because we started with multiplication, we were able to give students a lot of exposure and experience with activities and routines really early on, and help them make sense of what multiplication is before we started moving towards building more with facts. 

Pam Harris  02:42

And to be clear, it's not like you skipped place value, addition and subtraction. 

Kim Montague  02:45

No, no. No.

Pam Harris  02:45

It's just that you started with kind of rudimentary, a lot of skip counting, multiplication. Because you started with it, then you had the whole year to mess with it. I'm a little curious. Did you do the same thing in fourth and fifth grade? Like, let's start with it off the bat, give you lots of time, take the long view over the whole year? 

Kim Montague  03:03

Yeah. 

Pam Harris  03:03

Yeah. Cool. So let's talk about what you did do. Like what to do. What in the world then did you do all year long to help students learn relationships and develop their multiplicative thinking, so that the multiplication facts were a great byproduct?

Kim Montague  03:18

Well, so again, I started the year with multiplication. And if that's not an option, because your district or school has a mandated scope or sequence that you feel like you can't mess with, you can still introduce some multiplicative relationships in small doses throughout your week. And I think I've mentioned before that two thirds of the math time that I had was core content that we followed a scope and sequence. And then the other one third was a mixture of all kinds of things. And so if I wasn't able to help shape the order of material, I think I would have done some things more consistently in that one third chunk of time, because the amount of time we give kids to become reasoners, about facts really matters. 

Pam Harris  04:02

Absolutely. So order matters. When we think about the order of your sort of school year that matters. Order also matters when we think about how to build the facts, because we can base some facts off of others. So we want to think about the order. Like Kim just said, the order that you teach throughout the year, but also the order in which you work on the facts. So we often see a lot of teachers and some of the things that we talked about, that we're not crazy about in the last episode. So if you haven't listened to the last episode, we highly recommend that you go catch that, because a lot of those things that we don't recommend, and we see in a lot of places, typically start with like times one, times two, maybe times zero first, and then times one, times two, times three, times four, like they go in order.

Kim Montague  04:50

 Right. 

Pam Harris  04:50

But we suggest that it's much more beneficial to explore these relationships in maybe a different order. Like what are the relationships that would be helpful for students to think about? So, ready? Let's experience that for you listeners as learners. So we're gonna actually experience this a little bit. Here we go. 

Kim Montague  05:07

Okay, so today you're going to be the person. Right? I'm going to give you some problems. Can you just talk out your thinking a little bit? 

Pam Harris  05:17

Yeah, absolutely. 

Kim Montague  05:18

Okay. So today, you're going to think about 23s. 

Pam Harris  05:23

Hey, so since I'm talking about my thinking, I'm going to tell you I just picked up a pen, and I wrote down the number 23. So that's what's on my paper right now. 

Kim Montague  05:29

I grabbed a pencil.

Pam Harris  05:31

I'm a pen person, Kim's a pencil person. So good. 

Kim Montague  05:34

Okay, all right. 23. Alright, what is two times 23

Pam Harris  05:39

Two 23s is 46. Because I double the 20, double the three and that's 46. 

Kim Montague  05:45

Okay. 46. What is four times 23? 

Pam Harris  05:49

So I'm going to be lazy and just double two, the 46. So if I double 46, I'm gonna double 40 that's 80, double six is 12. So that's 92. I'm gonna check that by thinking about double 50. Double 50 is 100. But I need to take off the extra four twice. And that's 92. Yep. So 4 23s is 92. 

Kim Montague  06:10

Okay, that was pretty quick. So two 23s is 46. And double that for four 23s is double 46, which is 92. 

Pam Harris  06:19

Yep. 

Kim Montague  06:19

Okay. Let me ask you eight times 23. 

Pam Harris  06:22

Oh, nice, because I can work off the four. Okay, so if I know that four 23s is 92, then eight of them is going to be double that. And double 92, I'm just gonna double the 90 to get 180, and double 2 is 4. So 184. 

Kim Montague  06:34

Okay. Alright. What about 10 times 23? 

Pam Harris  06:39

Let's see, I know this thing in our place value system where I can think about 23 tens. And so that's 230. 23 tens. 

Kim Montague  06:45

Okay, 230. 

Pam Harris  06:48

I guess I should say, that's the commutative property that I just used. 

Kim Montague  06:52

Oh, yeah, because I asked you 10 times 23. But you really thought about 23 tens? 

Pam Harris  06:57

Yeah, so the commutative property. Okay. 

Kim Montague  06:59

Okay, so 10 times 23 was 230. What if I asked you five times 23? 

Pam Harris  07:04

Oh, nice order, because I can just totally work off the 10. So if ten 23s is 230, then five is half of 10. And so half of 230 is 115. 

Kim Montague  07:15

Right. So you just thought about 10 times 23?. 

Pam Harris  07:20

Yep. 

Kim Montague  07:20

And so what is nine times 23? 

Pam Harris  07:23

Oh, nice. Let's see. So that's just going to be 23 less than 230. And so 230 minus 20, is 210 minus three is 207. So it'd be 207. Is nine 23s. 23 less than ten 23s. 

Kim Montague  07:40

Nice. 

Pam Harris  07:41

Okay. 

Kim Montague  07:42

That's it. Well done!

Pam Harris  07:43

Cool, cool, cool. So let's look at those relationships that you just sort of emphasized. You asked me for two of them, to get four of them to get eight of them. And I could just sort of double double double. 

Kim Montague  07:57

Yeah.

Pam Harris  07:57

Y'all we want kids to know that they can find eight times anything! By double, double, double. And they can find four times anything just by double doubling. That's a powerful relationship that kids could use. And it helps them sort of begin to think about doubles. And that's a multiplicative way of thinking. Then you gave me 10, which if we have this community property, and we can learn the pattern in our base 10 system that we can think about that zero and how I'm thinking about 23 tens. Anyway, so 10 is easy to get right off the bat. And then you ask me five right after that. Oh, brilliant, because I can just cut it in half, since five is half of 10. Okay, totally cool. Then I don't know, listeners, if you noticed, but then Kim said: and ten again? In some way you asked me for 10 again.

Kim Montague  08:42

Yeah. 

Pam Harris  08:43

Which then pulls up that 10. And then you asked me for 9. So that's a way to sort of nudge: Oh, you have that 10, can you use that to help you think about nine? I mean, you didn't say alright now use the 10. 

Kim Montague  08:55

Right. 

Pam Harris  08:55

Here's what you do: step one, step two. You just like mentioned," Oh, yeah, what was that 10 again? Now nine?" And then I'm sort of nudged, the 10 is poking in my brain. Pinging is the word I want. Pinging in my brain. And so then I'm tempted to sort of use it to find the nine. What a great set of relationships. So let's continue to parse that out. Y'all Do I need to know my 23s? We're talking about single digit facts. These episodes we're working on right now is all about single digit facts or know your multiplication facts. Is 23 one of the facts that's on your list? Not typically, right? Typically, we ask kids to know the single digit facts or 10 by 10, or maybe 15 by 15, even though we think that's a little much. So 23 is not on that list, but it's about the relationships. So those relationships that Kim just had me use those are important. So can we use those same relationships to figure out single digit facts? Absolutely. We want kids to be able to think about sevens. They can think about eight times seven in the same way I just thought about eight times 23.  Can they double seven to get 14? I should have asked you that Kim. Kim, if double sevens 14, what's four times seven?

Kim Montague  10:09

Hmm, double 14, would give you 28. 

Pam Harris  10:12

And then what's eight times seven?

Kim Montague  10:14

Double again, right? Which is double 28 and that's 56. 

Pam Harris  10:18

Yeah. And if you say to yourself right now, "My kids can't double 28. I can't even double 28." Then what we recommend is that you start doubling with them, double yourself, like that's a thing to do is just to start doubling with kids. They get better and better at doubling, and then they can! Then that's a relationship that they can use. So let's just again, reiterate. So 2, 4, 8, great relationships. Ten to get to nine, or 10 to get to five. Great relationships. So there's one that we didn't do. Just for fun. We'll do it now. What if I wanted to ask something like: Kim, I'm going to ask you to think about 31s. 

Kim Montague  10:57

Oh, okay. 

Pam Harris  10:58

Again, not a number that's like typically a single digit factor, right? So we're going to kind of exemplify it with a bigger number and then back up to single digits. So 31. 31. What's three times 31? 

Kim Montague  11:11

Okay, three times 31. That's, I'm thinking about three times 30, which is 90, and three times one is three. So that's going to be 93. 

Pam Harris  11:20

Totally cool. And that wasn't too bad, right? You didn't have to double it and add one more, you could totally just jump right to three times 31. So if three times 31 is 93, three times 31 is 93. What's six times 31? 

Kim Montague  11:35

Ah, there's that doubling again, right? If three 31s is 93. And now I want six 31s? That's double the number of 31s. So I can double 90 and double three and get 186.

Pam Harris  11:49

Bam! Right? So you just like thought about six times 31 using three times 31. Now, could you have thought about six times 30? And six times one?

Kim Montague  11:59

Sure. But I already had the three groups.

Pam Harris  12:02

So why not use it right? And then we could kind of connect that to single-digit. So for example, Kim, keep going: What's three times seven? 

Kim Montague  12:12

Three times seven is 21. 

Pam Harris  12:14

Right? So then what's six times seven? 

Kim Montague  12:17

Ooh, that's a pretty commonly missed fact. So in fact, my son the other day was thinking about six times seven. And so three times seven is 21. I can double that to get 42. 

Pam Harris  12:30

And check out how easy 21 is to double, right? I mean, 21, if you can double it all, you can double 21. So that is my favorite, favorite relationship to get that most missed fact of 6 times 7. What was your kid, how was he thinking about it? 

Kim Montague  12:45

Actually, it wasn't doubling. He was messing with square numbers. And he was thinking about six times six, and added another six. 

Pam Harris  12:55

To get seven sixes. 

Kim Montague  12:56

Yeah.

Pam Harris  12:57

To get six times seven. So that's a fine relationship. And we would love kids to recognize that six times six is related to six times seven. Absolutely. We also want them to own this idea that they can double from three to get six. 

Kim Montague  13:09

Yeah. 

Pam Harris  13:10

Totally cool. Let's keep going. So if six times seven is 42, what's 12 times seven?

Kim Montague  13:17

Double again. So I know six sevens. So to get 12 sevens, then I can just double the 42 from the six sevens and get 84. 

Pam Harris  13:26

And now you have 12 times seven. So y'all, it can be really helpful to have these relationships. We're not suggesting that kids laboriously refigure these relationships, ad infinitum. The older they get, the more experienced they get. But the more experience they get with these relationships, they actually go quite quickly. Yeah, in fact, I'm going to suggest that the kids might run into 12 times seven and go, "Let's see." And just in their brains go, "I know it's twice forty- 84." And it could be that fast, it can be faster for them to go, "I know it's twice fourty - 84." Like they can just think about the six times seven to get the 12 times seven. Not only can that be a way for students to do that. That is the way that mathematicians pull up multiplication facts, they don't just cold remember. And often that is true that they don't just cold remember, that they think of some quick relationship that they know. They figure it, it doesn't bog them down in the work that they're doing, and they keep going. But even more importantly, they're building multiplicative relationships as they go, that that's why they're able to do it is because they own those relationships. 

Kim Montague  14:33

Can I bring up another point? We also think about language as well. And I think I've heard both of us alternate our language between saying things like 'two eights' and 'four eights' and then later alternated into saying 'two times eight' and 'four times eight'. 

Pam Harris  14:51

Mmhmm. 

Kim Montague  14:51

And we do a lot of work with doubling and then bring in the language and the recording of the times symbol, because kids can be super freaked out by the notation. Or having heard maybe that multiplication is hard. So instead, we advocate for having them experience multiplication, and then bring in the vocabulary and the symbols. 

Pam Harris  15:10

Yeah. In fact, when you bring that up, I just thought of a little guy I was working with the other day, we were just chatting about some mathy things. And he's young. And so we were doing some addition, subtraction stuff. And he looked at me with these big eyes, he goes, "But I can do multiplication!" Because like, it's this thing, right? I could do multiplication! And we had been working with some fairly big numbers. And he's like, "I know, two times four!" And I was like, "Okay, good for you, it's just two fours." You know, like, if he really understood that two times four wasn't this crazy thing, it was just two fours, I don't think he would have been quite as impressed with himself that he knew two times four. Because it was just sort of this outside crazy, big impressive thing. So we like to make it much more about the meaning of what's happening, then bringing the language like you said, and then bringing the notation last. And so the kids are like, "Oh, yeah, that's what we've been doing. And now this is what it looks like on paper? Okay, totally cool. No problem." And that fits, Kim, with what you just talked about with vocabulary just in time. We've talked about that before in the podcast, where we want to give vocabulary just in time when students need it. It's a great example of that. Hey, it's also reminding me really quick of a story. When my daughter, Abby, was in fifth grade - I'm pretty sure - she had to do some multiplication timed tests. And at that point, we were living within walking distance of the school, kinda. And so she wanted to walk to school. And I said, "I will walk to school with you. If," because then I had to walk back home, right, then walk back to school, and back home with her, so I was doing double the walking, I said, "If we can, like, do some modification facts stuff on the way." Because my kids every once a while, they're great at math, they mathematize with the best of them. But sometimes doing it with your mom, you know, is not like your ideal. 

Kim Montague  16:51

You don't say? 

Pam Harris  16:53

She's an artist, she wanted to talk about other things. And so she agreed. And so as we were walking, I would literally do kind of what you did. I would say. "Okay, let's talk about - and pick a random fact. Nines! Let's talk about nines. Hey, Abby, what's two times nine? What's four times nine? What's eight times nine? What was two times nine again? What's three times nine? Cool! What's 10 times nine? What's nine times nine? What was 10 times nine again? What's 11 times nine? What's 12 times nine?" And then other times I might have gone, "What's 10 times nine? What's five times nine? Well, five times nine is that what's six times nine?" 

Kim Montague  17:27

Right 

Pam Harris  17:28

"Well, if that's six times nine, what is seven times nine?" Or I might have gone, "What's two times nine? What's three times nine? What's six times nine? Oh cool, then I know you have six times nine, what's seven times nine?" Like I'm using those relationships, back and forth, in the air as we were walking. So parents, we suggest you can do this kind of thing in the car. Now to make it fun, have the kid talk about what they're thinking about and you share what you're thinking about. That makes it much more fun. Don't make it be this like, I'm just gonna drill you right now. It's not about drill. It's about using the relationships. It's about talking about the connections in your head. 

Kim Montague  18:00

Yeah, that's great. And I think it's really helpful to spend a lot of time talking about those relationships, instead of making it be something that you do once and then expect that they know. Right? 

Pam Harris  18:14

Well put. Yeah, good remember. It's not about, "Hey, we did your sixes in the car last Thursday!  You should know them!" No, no, no, no, no. Like, it's like building these relationships and then exposure and having lots of fun doing them often. 

Kim Montague  18:30

Yeah, sure. Okay, so today, we shared just a few ideas about how to work with students on their multiplication facts, right? But you're really going to want to listen to next week when we share the final episode in the series and give you some great routines of games and ways to continue building relationships for all students. 

Pam Harris  18:47

Yeah, you are not going to want to miss that one. So stay tuned for next week's episode. Hey, we promised you a download with some really helpful things. The download has helpful Problems Strings for single digit facts, like the strings that Kim and I did today live on the podcast. So if you'd like that handy dandy download, you can get it on the show notes. Or go to mathisfigureoutable.com/factsps. So mathisfigureoutable.com/factsps. 

Kim Montague  19:17

Yeah, we would absolutely love it if you join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evening, where we explore problems with the whole world.

Pam Harris  19:27

Please share with your friends and your colleagues and give us a rating and a review so that more people can find the podcast. So 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!