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

Kim  0:11  
And I'm Kim Montague, 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  0:19  
We invite you to consider that algorithms are amazing historic achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:34  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:41  
We invite you to join us to make math more figure-out-able. Ba dum tss. 

Kim  0:47  
Hey, there. 

Pam  0:48  
Okay, so I actually had a lot of that memorized because I had forgotten to scroll up.

Kim  0:53  
Good for you.

Pam  0:54  
I was like scrolling as I was like, "Okay, what is the beginning of this?" I was trying to get it down. 

Kim  0:57  
I mean, seriously, this is the 303rd episode. You would think. I mean, I know we switched at one point a couple years in, but you would think.

Pam  1:08  
We did write a new. Yeah, do you think we'll ever write another new beginning? Probably, right? 

Kim  1:12  
Oh, yeah, sure. 

Pam  1:13  
We'll do something new. Yeah. Alright, cool. So, Kim, you and I were chatting a little bit about what we were going to do in the podcast. And I shared a recent experience with you. And we thought we would bring it to our podcast audience because there's some interesting things that happened and we wanted to parse out a little bit what you thought about it, and what I think, and kind of where we go from here. So, I was doing a workshop. And I've done several in-person workshops lately. I don't do as many of those lately or as many. Yeah, anyway. Blah, blah, blah. I was doing an in-person workshop with some wonderful educators. We had kind of range the gamut. We had everything from... They were mostly secondary. Sixth grade up through... We had calculus teachers in the room. But we also had an interventionist, GT specialist. We had the assistant superintendent for curriculum instruction. And I'm not actually quite sure, but we had this social studies expert. So, you know, quite the range of people in the room. It was a great, great group. Had a fantastic day. In the day, I started the day like I often do, telling the story about my kid when I was talking to my kids' teachers, and they said, "Hey, you know, we've heard we don't want to give kids math anxiety, so we should stop giving kids time fact math tests. So, we stopped giving them time fact math tests because we don't want to give them math anxiety. But now we're noticing kids don't know their facts anymore. Is that important?" And I was like fussing with that. I'm wondering, is that important? I wonder. I wonder, does my kid know his multiplication facts? I go home. I ask him. Just pulled one at random. Do you know your fives? He's like, "No." I'm like, "Ah! What?! You don't know your multiplication facts." You know, sort of really generalize if he doesn't know one sort of factor of multiplication facts, then he doesn't know any of them. And I'm panicking. Yeah, I'm panicking. And he says, "Mom, mom, mom! You don't have to know your 5s if you know your 10s!" And in that moment, there was this mind blown, you know, of me thinking, "Wait. What does it mean to know your multiplication facts? Do kids need to have them memorized? Or would I rather have this confidence and relationship owning that you don't have to know your 5s if you know your 10s because you just multiply anything by 10 and cut it in half to get times 5." And he even did it with times, you know, double-digit numbers. And so, it was this mind blown moment of me thinking, "Wait, what does it mean for kids to know their facts? You know, what's more important?" And that's the main kind of message of that opening story. And I ask participants what's resonating for you? What do you have a question about? And a couple people said that they were grappling, kind of. It sounded like they were grappling with the same thing. Like, you know, we have this feeling that kids need to know their multiplication facts. But wait, we're kind of interested in what your kid was doing. And blah, blah, blah. So, later. We went on. We did a bunch of Problem Strings. We're doing a lot of work. And at one point, we're doing a Problem String, and one of the participants raised their hand and said, "But wait. This stuff that we're doing right here. How's that going to help kids know their facts? Weren't you worried that kids don't know their facts?" And I was like, "Um, yes. But remember, there was this epiphany that, you know, we're grappling with. What does it mean for kids to know their facts?" And it was just interesting that as we were doing these Problem Strings to really build relationships, so that strategies became natural outcomes. You know, at least one of the participants was still grappling with the work that you're doing. I think this person clearly heard at the beginning, "Kids need to know their facts." Maybe didn't hear so much the conundrum. The, what, the grappling with what does that mean? And how much? And does it really mean they need to? Yeah. And so, they were kind of bringing it back up. "Well, wait. Wait, this work that we've just been doing. This isn't helping kids know their facts." And I was kind of like, "Yeah, but what is more important?" Yeah, so that went on. It kind of came up again, kind of. I was trying... Kim, I try really hard in professional learning situations, I think you do too, to not just tell people stuff, and then try to prove it, but to try to engage people in experiences, so they come to the realization themselves, right? Like, I raise some ideas. We do a bunch of math. And they start to go, "Wow, we can really see how it's more important that kids build these relationships because then they're going to own... I think you've said it this way. They're going to own enough facts, that then they can build from there to ones they don't know, including ones that we don't traditionally think that they should know. Anyway, that's going to come up a little bit more in my story here. So, later in the day, I did a factor to subtract strategy. Now, we've done that on the podcast before, so maybe we can link in the show notes where we've done that if you want to go listen to that. But I'll replay it a little bit enough here that we kind of get a feel for what's going on. So, for example, I might give a problem like 77 minus 50. And just really quickly throw up a number line. 77 minus 50. And what is that? Is that 27? Help me, Kim. I'm doing live here. And then if 77 minus 50 is 27, then the next question is 77 minus 49. And obviously... Maybe I shouldn't say "obviously". But because I've put up that 77 minus 50, a lot of people will try to use 77 minus 50 to help them to think about 77 minus 49. Which is fine. And we talk about that Over subtraction strategy. But then there's this punch where I'm also... I'm trying to accomplish a couple of things in this Problem String. Then, the next problem is what is 11 times 7 minus 7 times 7? And I'll even say something like, "Completely unrelated. You know, totally unrelated to the problems before." And then I'll smile a little bit, and I'll say, "Was anybody a little lazy? I mean, efficient. Were you acting, you know, doing a mathematical behavior of where you're trying to be efficient?" And usually somebody, and in this day somebody said, "Well, yeah. Just once I started to think about 11 times 7, that's 77 minus 7 times 7, that's 49. I remembered those numbers." Like, it's like, "Oh, it's almost like we've just solved this problem." Well, sure enough, it was the problem before. And so, then they're like, "Yeah, I was a little bit lazy. I just said, since we've already solved that problem is 28, then I'm not even going to do any more work. I'm just going to say that eleven 7s minus seven 7s is 28." And I'll say, "Great, great. That's a great strategy." And I'll try to pull that one out first. And I did. And then I'll say, "But did anybody not do that? Did anybody not just say, oh, this is, like, the problem before, so I'm just going to take the answer from the problem before. Did anybody actually think about eleven 7s minus seven 7s? If you know eleven 7s minus seven 7s." And, usually, in this day, sure enough, there were a couple people that kind of grinned. And I was like, "You're grinning. Like, tell me about that." They're like, "Well, yeah. Like, eleven 7s." They said, "I didn't even think about the subtraction problem before. I just saw 11 things minus 7 things, and that's 4 of those things. So, four 7s. And I know four 7s is 28." Bam. And that's kind of how they solved the problem. Often, when this happens, and this happened on that day, a lot of people in the room kind of raised their eyebrows. And they were like, "Huh. That didn't occur to me." And I'm like, "That's okay. It's alright. Cool. Here, next problem." Then, I give them another problem like 81 minus 40. We do the number line for that. Then,we do 81 minus 36. We talk about can you use 81 minus 40 to help you think about 81. Again, I'm ramming through this, Kim, because you can go listen. 

Kim  8:40  
Yeah, yeah.

Pam  8:40  
If you haven't heard this Problem String, go listen to the Problem String. Could use 81 minus 40 to help you think about 81 minus 36. Often, people are smiling who hadn't seen the relationship before. They're thinking about this Over subtraction strategy. They're, you know like, making these jumps and stuff. It's great. And there's also a couple people that are kind of winking at me a little bit. And sure enough, there was these two guys that were sitting at one table. And I was like, "Why are you smiling?" And the one guy. I think his name is Chris, maybe. He said, "Well, 81 that's like nine 9s. And 36, that's like four 9s. So, nine 9s minus four 9s is five 9s. And five 9s is 45." And I'm like, "That's actually how you solve the subtraction problem?" And he's kind of like, "Yeah." And a lot of people in the room are like, "What?!" And he's like, "Yeah, it's just like 9s are popping out at me. Nine 9s minus four 9s is five 9s. And so, five 9s is 45. And so, I think the answer to that subtraction problem is 5 times 9 is 45." So, then we did a couple of more problems and everything. So, as we're doing this Problem String, that same table that kind of pushed back earlier, "But wait, don't we want kids to know their facts?" again, said... Well, this time kind of almost triumphantly. It was kind of interesting. They were like, "See! See! See how important it is that you know your facts!" And, Kim, I was looking at the board, and I was like, "Say more about that." And they said, "Well, Pam, in order to solve 77 minus 49 using 11 times 7 times 7 times 7, you had to recognize that 77 is 11 times 7. And like you had to recognize 7s. And to solve 81 minus 36, it had to ping you for 9s. And so, see, we have to teach kids multiplication facts." What's interesting then... So, I kind of let that push back. I kind of raised it. I was kind of like, "Okay, that's interesting that you're sort of seeing that." And then I threw out the problem 144 minus 96. And I said, "What do you got for 144 minus 96? What could be a way that you could solve that problem?" And several people around the room who'd never messed with Over subtraction started thinking about 144 minus 100 and going from there. So, I'm circulating around. I'm asking people. But also several people heard 144, and they thought, "Ooh, that's twelve 12s."

Kim  10:59  
Mmhm.

Pam  11:00  
And so, as I'm circulating, I found a group, and I called on them, and they said, "Well, we can think about 144 as twelve 12s. And then 96 as..." And we kind of had to think about that, because a couple of them were like, "We never memorized our 12s." And I was like, "Well, could you think about 96?" And they're like, "Yeah, we thought about 96 as a few less 12s." And sure enough, it was just eight 12s. There was a couple different ways that they figured out the factors of 96. Noteworthy. They didn't know that 96 was eight 12s in the moment, but they were able to reason quick enough that it didn't bog them down in the work they were doing. And so, then they shared. So, then it's twelve 12s minus eight 12s. That's four 12s. And four 12s, we know is 48, and so the answer to the 144 minus 96 is four 12s. And I said, "Oh, that's fantastic! Like, I love how multiplication is pinging for you. This is great, great. I wonder. Did anybody have any other facts? Any other? Not 12s. Anybody have anything besides 12s that pinged for you from 144 minus 96?" I let them go. I let them loose. I circulate around a little bit. I'm watching. Kim, it was fascinating. The group that was really focused on, "Kids need to know their facts." Like, they kept like, "See!  The work you're doing isn't helping kids memorize their facts." When I invited them to think about other factors that pinged for them for 144 minus 96, they had nothing. Now, I'm not blaming them or ridiculing them at all. It's just noteworthy that they were so insistent that how important it is that kids memorize their facts because they memorized their facts. So, for example, at some point, I said, "Hey, what else do you got?" And somebody said, "Well, I could think about 24s. And I was like, "Tell me about that." And they said, "Well, if it's twelve 12s to make 144..." In fact, Kim, maybe I should ask you. How are you thinking about, if you know twelve 12s makes 144, how does that help you think about how many 24s make 144?

Kim  12:58  
So, if I have twelve 12s, then I could have twenty-four 6s because I have twice as many where the group size is half as big.

Pam  13:11  
Nice, nice. 

Kim  13:12  
Say that again. 

Pam  13:14  
Yeah. Or similarly, I could have half as many.

Kim  13:18  
Mmhm.

Pam  13:19  
So, 6 groups where the groups are twice as big. 24. Now I have six 24s. And then stay there because I want to talk about 24s. 

Kim  13:29  
Yep. 

Pam  13:30  
So, or could really go either way. I'd love your way. And then this other way thinking about twenty-four 6s or thinking about six 24s? Could you similarly think about eight 12s? If I know that I have eight 12s, how many 4s do I have if I have eight 12s?

Kim  13:45  
Four 24s. Did you say eight 12s?

Pam  13:49  
Yeah, eight 12s.

Kim  13:49  
Yeah. 

Pam  13:50  
Why do I have four 24s if I have eight 12s?

Kim  13:52  
Because I have half as many groups that are twice as big.

Pam  13:56  
Kim, do you build that kind of intuition rote-memorizing single-digit facts?

Kim  14:03  
No, no.

Pam  14:03  
You don't, right? How do you build that kind of intuition? You build that kind of intuition when you're trying to help kids really own multiplicative reasoning. And so, you're in and out of these relationships. So, now we end up with six 24s minus four 24s. Is that two 24s? Hey, and two 24s, that's also 48.

Kim  14:22  
Yeah. 

Pam  14:22  
So, in that moment, guess what that table shot back with? That table shot back with, "Yeah, yeah, yeah. But we don't expect kids to know their 24s. You know, in the reasonable things that we..." You know, they didn't even say "reasonable." What they said was, "The ones that kids should know." And I was like, "I mean, like, so what's magic about the multiplication table up to 12s? Because we just had teachers right here that admitted they don't know their 12s. And Oh, bless his heart. That's when one of the calculus teachers raised his hands and says, "Yeah, I don't actually know my 12s." And I was like, "So, that's noteworthy. We got a calculus teacher in the room that, you know, is fully admitting doesn't know his 12s. Oh, no. Oh, no. He's going to fail. He's not going to be successful. Or are there more important basics? Are there more important things that we can help kids?" Like, Kim, the reason I said "basics" is you and I have been talking lately about how there's some folks out there, some people out there that are saying, "It's really important that kids know their basics before they can do any kind of reasoning with mathematics." And we're going to maybe redefine what those basics are. And it's things like we just did, where if you know twelve 12s, can that help you think about how many 24s you have? Or, Kim, can you go from there? So, I'm just going to see like what else you got? We got twelve 12s minus eight 12s. We got six 24s minus four 24s. Can you think of any other factors that we could pull out of 144 minus 96?

Kim  15:43  
Yeah, you can do three 48s.

Pam  15:47  
Okay, so we got three 48s minus?

Kim  15:50  
Is that two 48s?

Pam  15:52  
I don't know. Is it?

Kim  15:54  
I didn't write mine down, so I wish I would have.

Pam  15:56  
Three 48s minus 2.

Kim  15:57  
this whole time. Yea I think its... (unclear)

Pam and Kim  15:59  
So that would be one 48.

Kim  16:01  
Yep, mmhm.

Pam  16:03  
Oh, I like what you just did, but I also just checked to see if we got the right answer. Like, three 48s minus two 48s is one 48. And, sure enough, the answer the whole time has been 48.

Kim  16:11  
Yeah. 

Pam  16:11  
So, now I do wish you'd been writing it down. And this is one of those moments where the visualizing of me recording thinking would be so important. 

Kim  16:19  
Mmhm.

Pam  16:19  
So, once I've got twelve 12s minus eight 12s, and six 24s minus four 24s. Yeah, can you come up with any more? Now, that I know you don't have them written down, that's much harder to do. 

Kim  16:30  
Yeah.

Pam  16:32  
I will say somebody came up with 2s, how many 2s they had. 

Kim  16:37  
Yeah.

Pam  16:40  
I know I could do 96 is fourty-eight 2s. What's 144 cut in half? Is that 70...

Kim  16:46  
72.

Pam  16:47  
2. So, we'd have seventy-two 2s, minus fourty-eight 2s. And that's when somebody said, "Why would you do that?" And somebody goes, "Yeah, you wouldn't. But you could consider it. Like, we now have enough sort of relationships happening that we consider it." And somebody said, "Well, if we know seventy-two 2s, and we don't want to do 72 minus 48. Like, that doesn't make us happy. But once we have the seventy-two 2s minus fourty-eight 2s written down, could that help us think about the number of fours? Is that thirty-six 4s minus... Can you see how I'm Doubling and Halving here? Twenty-four 4s.

Kim  17:20  
Mmhm..

Pam  17:20  
Oh, well, that's just twelve 4s. Oh, bam. We're back to 12 times 4 which is 48. Y'all, in that moment, it was interesting to watch that table that was so like set that kids need to rote memorize their facts, go, "Yeah, but these aren't reasonable for kids." And I'm like, "Wait, wait. We're not expecting kids to rote memorize any of these. What we're doing is we're saying, If we build multiplicative reasoning, look how they are now reasoning multiplicatively. Like, and not as an end product, but we're building it here. Like, we're continuing to have kids sort of build from here." Yeah. 

Kim  17:51  
So, what I love about the string... Thank you for sharing that story. And I wrote several notes down that I wanted to come back to, but I didn't interrupt you. So, what I love about this kind of string is because you're sharing problems like 81 minus 36 and 77 minus 49 that you could think of the quote, unquote, "basic fact". But when you land with a problem like 144 minus 96, there's so much more richness to it that keeps kids playing and looking for more. And I'm glad that you said that they're building because you could have stopped at 12 times 12 minus 8 times 12 and just said that those are two facts you're supposed to know. But you're deepening their understanding as they're looking for more. And you're building. You probably already said this well. But you're building their multiplicative reasoning. And I'm thinking about the people that you were working with, and I wonder if maybe they've only thought about facts, and they think about it like, "Okay, I'm supposed to know this series of facts." And they picture the problem. They picture the 9 times 4 and 7 times 4, and they think this type of problem, they have to go through all the facts and see which one lands at 81. Like, I wonder if they saw 81, and they thought, "Okay, what's the Rolodex of problems? Which one hits on 81?"

Pam  19:21  
That I've memorized, yeah, that 81. Mmhm, mmhm.

Kim  19:25  
Where you and I would think, "What do I know about 81?" Like, we're picturing that. And I think we're calling this like a product approach because when we picture 81, we think, "What do I know about this problem?" We focus on deeply understanding a number. So, it reminds me. You know, you used to joke. Like, when we played, I Have, You Need a ton early on with teachers. You would say to them, when I wake you from a dead sleep, and I say, "What do you know about 64? And then you rattle off some examples. So, like what would you want somebody to know about 64?

Pam  20:00  
Yeah. Well, and I would usually ask people, and they would come up with things like 60 plus 4, 70 minus 6, 8 times 8. And I would say, "Keep going." Yep. And then they would say, "Okay, how about 2 times 32, or 16 times 4. You know, and we kind of had to get going a little bit to get the juices sort of flowing, but 8^2. Like, we really want to have a lot of relationships, including 64 plus what is 100? We want to also have that 36 coming out. Yeah, what do we know about the number? Yeah, that's nicely said. That's a good connection. Yeah. 

Kim  20:31  
Well, and the more that people know about a number, the more flexibly we can move around that number. The more that we see a number, and we start to rattle off all of the things. So, like, when you see 144, if you only think 12 times 12, maybe we'd wonder if it's because you've only experienced it by knowing the fact 12 times 12. And we would want richer experiences for somebody. 

Pam  20:59  
Yeah, like you and I were just Doubling and Halving and thinking about how can I go from what I know to something else? Which we build when we're not just focused on rote memorizing a small set of facts.

Kim  21:09  
Yeah.

Pam  21:09  
And then we get a whole lot more. Hey, Kim. Try just one more with me. 

Kim  21:12  
Yeah. 

Pam  21:13  
What if I were to ask you 72 minus 48? What comes to mind for you if you were going to solve that problem?

Kim  21:22  
I think I'd think about nine 8s minus six 8s because I know those two have eights. Yep. 

Pam  21:29  
Nice. Nice. And so, you could also have thought about 72 minus 50, if we were doing the Over thing. But cool.

Kim  21:36  
Mmhm, yeah. 

Pam  21:36  
Do you got any other factors that pop out at you?

Kim  21:41  
Let me think. They both have 6s. 

Pam  21:46  
Mmm, mmhm.

Kim  21:47  
So, is that twelve 6s minus eight 6s?

Pam  21:51  
Nice. 12 minus 8 is four 6s. So, that's also 24. I like it. I like it.

Kim  21:56  
Yeah, they're both even, so they have 2s as well.

Pam  21:59  
Yep, yep. And I actually doubled and halved and got eighteen 4s minus twelve 4s. So, that would be six 4s.

Kim  22:08  
Yeah.

Pam  22:08  
And then we're kind of back to the four 6s. Yeah.

Kim  22:12  
This could be such a fun activity to do in class. It's much more meaningful than a timed fact thing. Like.

Pam  22:19  
I'll bring up... Our friends from Innovamat have... They've coined a phrase, what they call "productive practice".

Kim  22:25  
Yeah.

Pam  22:26  
I think this is an example of "productive practice" where we're getting kids to really do a lot of multiplication facts, but in a way that it's kind of this product approach, and the relationships between the factors. And then, if I could just mention. So, I'll just take the one that you said 9 times 8 minus 6 times 8? Could I then say to say middle school kids, "Hey, it's almost like you're saying if you have nine 8s minus six 8s or 9 things," and then I write down 9x, "minus 6 things," and I write down 6x. "You're saying that that's like 3 of those things," and I write down 3x. Now, we're helping kids reify an algebraic term, that they can look at 9x and know that it's 9 of those x's. And minus 6x. 6 of those x's. And now, they're not looking at 9x and thinking 9 plus x because they have this experience that we've helped them build before, which just leads... It could even lead to 9 times the quantity x minus 2, minus 6 times the quantity x minus 2? Well, if I got 9 times a thing minus 6 times a thing, doesn't matter what the thing is. It could be x minus 2, but I'm going to have 3 of those things. 3 times the quantity x minus 2.

Kim  23:36  
Yeah.

Pam  23:36  
Like, there's a lot of algebraic thinking that can come out of us really helping kids build these relationships. 

Kim  23:44  
Can I say one more thing that I have in my mind? 

Pam  23:47  
I mean, I guess. I'm just kidding.

Kim  23:51  
You know, I think sometimes the argument feels like it's either memorize your facts or don't. And I don't think that that's the case. So.

Pam  23:59  
No! 

Kim  24:00  
When you think kids need to memorize their facts the way that we would describe rotely, then we know sometimes that the result of that is either they know or they don't. And they say, "Like, I forgot," and they and they don't have a way to do anything about it. But I think what we're suggesting is that... 

Pam  24:16  
And a lot of kids get math anxiety along with that.

Kim  24:19  
Yeah, yeah. Yeah. 

Pam  24:20  
Mmhm.

Kim  24:20  
So, I think what we're suggesting is that when you build relationships, and you focus on those relationships, you also can build fluency. We would want you to do work to build fluency. But in that work, you're also building a deeper understanding of number in the way that we're suggesting a product approach helps kids. And so then, as people will sometimes forget, they can go back to the relationships. They can go back to the facts that they do remember. And it's not just this idea that like I forgot that one, and I don't have any way to do anything about it. So.

Pam  24:57  
I'm just going to have to guess or type in the calculator. Mmhm.

Pam and Kim  25:00  
Yeah, yeah.

Pam  25:01  
Nice. That's nicely said. Yes, it's not that we don't want kids to know their facts. We want them to more than know their facts.

Kim  25:07  
Right, right.

Pam  25:09  
Nicely said. Y'all, thanks for joining us today. We love having you as part of the Math is Figure-Out-Able movement. Thanks for tuning in and teaching more and more real math. Let's keep spreading the word that Math is Figure-Out-Able!