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  0:10  
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  0:18  
Because we know that algorithms are amazing human achievements, but they are terrible teaching tools. Mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:31  
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:38  
Y'all, thanks for joining us to make math more figure-out-able.

Kim  0:43  
In this episode, we're taking a look back at some episodes where we've chatted about important topics in 3-5 classrooms. We're highlighting one of the most sophisticated multiplication strategies based on equivalence, a model that you cannot do without in grades three and up, and some discussion about what we think about multiplication facts. Enjoy.

Pam  1:07  
Okay. Alright, first problem. 15 times 18. Have fun with that one because there's so many nice things you can do.

Kim  1:13  
Super. Oh, my gosh. You don't have anything in mind? I can do what I want?

Pam  1:16  
Yeah, do what you want.

Kim  1:18  
Okay, I'm going to go with 10 times 18, which is 180. And then I still need 5 times 18, and I know Five is Half of Ten. So, if ten 18s was 180, then five 18s is going to be 90, so together that is 270.

Pam  1:39  
Bam. Nice, Smart Partial Product. Nice using Five is Half of Ten. Cool. So, 15 times 18, you're saying is 270.

Kim  1:47  
Indeed.

Pam  1:47  
Next problem. What is 9 times 30.

Kim  1:52  
9 times 3 is 27 times 10 is 270.

Pam  1:57  
So, 9 times 30 is 270.

Kim  1:59  
Mmhm. 

Pam  2:00  
So, we now have two problems that have the same product. And I know you guys are listening, so I'll just say it again. We had 15 times 18 was 270. You just said 9 times 30 is also 270.

Kim  2:10  
Yeah.

Pam  2:10  
Kim, are those two problems related? If I were to draw... So, you mentioned...

Kim  2:16  
I'm literally sketching, as you say, "If I were to draw..."

Pam  2:18  
Yeah, so you mentioned you asked that kid in class to talk about the model.

Kim  2:22  
Yep.

Pam  2:23  
So, I have a 15 by 18 on my paper. If I were to draw a 9 by 30. Based on that 15 by 18. Could you help me? What would the 9 by 30 look like?

Kim  2:31  
Sure. So, I drew...  it's like kind of squarish, but a little bit longer on the 18 side. So, 15 down and 18 across is what I have. Is that what you have?

Pam  2:41  
Yes.

Kim  2:42  
So, across the top I had 18, but now I only want 9 across the top, so I'm cutting that 18 in half. I just kind of drew a dotted line down mine, cutting that 18 in half, so it's like a 9 and a 9 by 15. And I'm going to move that second half of that squarish one. It's not square, but you know what I mean. And I'm moving it down, so I'm making it twice as long and half as wide. My 9 by 30 is half as wide and twice as long as the 15 by 18.

Pam  3:16  
So, you actually just created a 30 by 9.

Kim  3:19  
Yeah.

Pam  3:20  
Yeah, even though I said 9 by 30, so maybe I shouldn't have asked you. I should have just told you. Let me tell you what I would have put on the board. So, I loved your explanation. It makes a lot of sense. You cut the 15 by 18 in half, you moved half of it down.

Kim  3:34  
Mmhm. 

Pam  3:35  
Created a long skinny rectangle. A tall skinny rectangle.

Kim  3:38  
You want me to make it be 9 by 30? Is that what you...

Pam  3:40  
Well, if I was running the Problem String, I would have drawn a 9 by 30, and I would have said, "So, how does the 9 relate to the 15?" You would have said, "Well, a little bit more than half." I don't know. And then the 30 would have been, what, not quite twice as long as 18. 

Kim  3:56  
Mmhm.

Pam  3:57  
So, you would have had this long, skinny but horizontal 9 by 30. Is that right?

Kim  4:03  
Yep.

Pam  4:03  
And then I would have said, Alright, so we're agreeing that they both have the same area of 270. Can you make that fit?

Kim  4:11  
Yeah.

Pam  4:12  
So, would you mind describing from the 9 by 30 that that horizontal flat-ish rectangle? How would you move from it to the 15 by 18?

Kim  4:22  
So, in that case, I would rotate it. 

Pam  4:26  
To create the one that you ended up with.

Kim  4:27  
Yeah.

Pam  4:28  
Yeah. And then you would have kind of that 30 by 9, and then you could have done the same kind of cut in half thing.

Kim  4:33  
Mmhm, mmhm.

Pam  4:34  
So, that seems like important that if you cut one of the dimensions in half and slide that bit of area down to make a long skinny one, which then, in effect, doubles the other dimension, you didn't lose any area. Huh. And we ended up with the same product. I wonder if that happens all the time. That's what I would say to students, yeah?

Kim  4:57  
Mmhm.

Pam  4:57  
Next problem. How about if I asked you 4.5, 4 and a 1/2, times 60.

Kim  5:04  
So, I'm looking at my previous problem, and I'm noticing that 9 is twice as much as 4.5 So, I'm sketching, and so right below my 9 by 30 that you had me draw, I'm drawing 4.5. But it's only half as long on the 4.5 side, so I'm going to actually make it be 60 long. And it kind of is off my paper, so I drew some dots. But a 4.5 by 60 is skinnier than the 9 and longer than the 30.

Pam  5:37  
And skinnier by how much?

Kim  5:38  
Half.

Pam  5:39  
And longer by?

Kim  5:40  
Double.

Pam  5:41  
Double. Cool, so you kind of halved one dimension, doubled the other. What is... Did you ever find 4.5 times 60?

Kim  5:48  
Oh, I didn't. It's also 270.

Pam  5:52  
And how do you know it's 270?

Kim  5:55  
Because if I cut that 9 by 30 into 2 pieces to make it be like a 4.5 by 30 and a 4.5 by 30, I could take that bottom 4.5 by 30 and slide it to create a 4.5 by 60. So, I didn't lose any area.

Pam  6:11  
So, we have an equivalent problem. Huh.

Kim  6:14  
Yeah.

Pam  6:14  
It seems like you've created equivalent problems all the way along. You've got 15 times 18 is equivalent to 9 times 30 is equivalent to 4.5 times 60 is equivalent to 270. So, 3 equivalent problems where you're kind of talking about cutting one dimension in half, doubling the other dimension, and the area stay the same, therefore the product stay the same. That seems kind of helpful.

Kim  6:38  
Mmhm.

Pam  6:38  
I wonder if you could do something like that for a problem like 24 times 3.5.

Kim  6:46  
Yeah, I was trying to make it relate for just a second. I...

Pam  6:50  
It might. It might not, right?

Kim  6:51  
Yeah. I'm... Okay, I don't really love the 3.5, so I'm going to double that one to make it 7.

Pam  6:59  
Okay.

Kim  7:00  
And so, that I can maintain the same area, I'm going to halve the 24 to make that 12.

Pam  7:07  
Okay.

Kim  7:09  
And so that is 84.

Pam  7:12  
Do you just know 12 times 7?

Kim  7:13  
Um... Yeah, I do.

Pam  7:15  
That's okay.

Kim  7:17  
Why do I say that like it's a disappointment? Yeah, sorry.

Pam  7:21  
Because you want to be able to talk about how you think about it, and you're like, "Darn it, I just know that one." Hey, y'all, just take a minute. Just take a minute to appreciate that that's math-ing. It's math-ing to say to yourself, "Ah, crud. I wish I had a way. I wish I had a reason right now to tell you how I'm thinking about that, and darn it, I just know that one." Wouldn't you love it if that was the result in your classroom, where kids instead of go... Like, how often do we hear them go, "I just know that one." And you're like, "Yeah, but tell me how you're thinking about it." "No, I just know that one." Instead we're like, "Oh, crud! I just know that one! Oh, man! That stinks, Gah, I wish I didn't just know that." I think that's awesome. Okay, cool. So, you could double one of the dimensions, halve the other. That you're saying that you could like take that area, slide it over, create a new rectangle that would be a 12 by 7. If you didn't know 12 by 7, which I'll just be honest, I have to refigure every time.

Kim  8:16  
Oh, okay.

Pam  8:17  
I know. Yeah, so I often just think about 10 times 7. And as soon as I have... So, like 12 times 7 is like 10 times 7 plus 2 times 7. But I don't do that much work. I literally think about 10 times 7, and I go 70, 84.

Kim  8:30  
Yeah.

Pam  8:30  
Like I just don't even think about that I need 2 more or whatever. It tells me I know that I'm going to be 60 for 12s, that I'm 60, 72, 84, 96, and I can just never remember which one's which. 

Kim  8:42  
Yeah.

Pam  8:42  
And so...

Kim  8:42  
Oh, that's me for 8 plus 5. Like, I think we all have... I, for whatever reason, 8 plus 5, I'm like, is that 12? 13? 14? I don't... But I go back to the fact that I know 8 and 4 is 12. For whatever reason, that was my favorite fact as a kid. So, then I'm like, "Oh, it's 12, 13."

Pam  9:03  
So, that 8 plus 5 has to be different. No, you're reminding me, actually. So, I was just in Oklahoma a couple weeks ago. Few weeks ago? I don't know how long ago. Anyway, super group of people. And, Cathy, the calculus teacher. When I said, "Does anybody not know 8 times 7? And, you know, you can admit it," whatever. And I always love it when it's the calculus teacher that raises their hand and goes, "Yeah, I don't know it." And she goes, "Yeah, I just can never remember if it's 54 or 56." 

Kim  9:27  
Yeah. 

Pam  9:27  
So, she refigures. I can't remember exactly what she does. But I think she refigures 9 times 6, and then says, "Oh, it's not that, so it's...

Kim  9:38  
Like, that's the 54.

Pam  9:40  
Yeah, she goes, "Oh, that's 54, so therefore it's 56." That was her. I think, Cathy, hopefully I just gave you credit there.

Kim  9:47  
I want lots of adults to admit that.

Pam  9:49  
I mean.

Kim  9:50  
I want lots of adults to say, "You know what? Like, I have to refigure one in a mathy way, and it's okay!"

Pam  9:56  
And it's more than okay because what we're saying is don't fill your brain with that trivia. Fill your brain with relationships.

Kim  10:02  
Yeah.

Pam  10:03  
So, that you can refigure it in an efficient enough manner, that it doesn't bog you down in the work you're doing. And at the same time, because you are using relationships, your brain is more developed and you can do more. And all the things. Yeah. Okay, cool.

Kim  10:15  
Back to the Problem String.

Pam  10:16  
Next problem. How about 32 times 12.5. I mean, Kim. Now, I'm giving you a 2 by 3 multiplication with decimals.

Kim  10:27  
Yep. So, I, again, don't love the 12.5. So, I want that to be 25, so I'm going to create the problem 16 by 25 by halving the 32 and doubling the 12.5.

Pam  10:46  
Okay, 16 times 25.

Kim  10:47  
Mmhm.

Pam  10:47  
Yeah.

Kim  10:48  
And I actually know that one. I have done enough work with quarters. But.

Pam  10:55  
Sure wish I knew something about quarters. I'm quoting Kim, by the way there. We have a video of Kim working with fourth grade kids, and this kid says something about 25, so she goes, "Sure wish I knew something... Or, no. You said something like, "What do you think about?" He goes, "Quarters." And you go, "Sure wish I knew something about quarters." Anyway, sorry, keep going.

Kim  11:12  
So, if at that point I didn't know something about 16 by 25, that's still a really nice problem because if you double and halve again, then you get 8 by 50. And you could even do it again and get 4 by 100.

Pam  11:28  
So, you're saying 4 times 100 is 400. And 8 times 50 is 400. And 16 times 25 is 400. Therefore, 32 times 12.5 is also 400?

Kim  11:39  
Yeah. Yep.

Pam  11:40  
So, you created lots of equivalent problems to solve that sucker, even though you probably could have only created one. 

We love ratio tables. Give us a high five and happy clap for ratio tables. Totally love them as tools. So, let's be clear. What is a ratio table? If you're like me and Kim, we never saw ratio tables as students, especially as tools for solving problems. They just didn't even exist. I dealt with a lot of problems in high school to do functions and functional relationships and all that. But what is a ratio table? And why might we be interested in using it to mathematize multiplication and division problems? So, a ratio table is a paired number table, but it's a special paired number table but it's a special paired number table where all of the ratios are equivalent, where all of the entries form ratios, and those ratios are equivalent. So, what does that mean in layman's terms? For example, if I had, say, the scenario where I had 27 sticks of gum in a pack. We like gum. We chew gum. You can picture a pack of gum. It has 27 sticks. Random, kind of. In that pack, then I have 27 sticks of gum in a pack. I might have a table that says 1 pack to 27 sticks is equivalent to 2 packs would have, double that, 54 sticks of gum. And all of those succeeding ratios. Like, I might have 10 packs would have 270 sticks of gum. So, it's the ratio of 1 to 27, 2 to 54, 10 to 270, could be 20 to 540. Like, all of those ratios are equivalent. And if those ratios are equivalent, then that is a special paired number table that we call a ratio table. So, you might find it interesting that a mathematician who works with Illustrative Math, Bill McCallum, wrote a fine blog post the other day called Ratio Tables are not Elementary. Which might lead you to believe that maybe we shouldn't use them in elementary school. However, we would actually agree with him that the way he describes a particular use of tables in grades three, yeah, would go away and isn't particularly helpful for much. We would agree with him in there. However, we conceive of ratio tables in the realm of multiplicative reasoning quite differently, differently than just a list of, say, single-digit facts. So, that could be a ratio table. I could have something that looks like 1 to 7, 2 to 14, 3 to 21, 4 to 28. And it can kind of be a list of what some people call multiplication tables. That would be like the table for 7s. That's a kind of limited use of ratio tables. But we offer an alternative view of how ratio tables can be used in developing and using multiplicative reasoning. 

I posit that ratio tables can be used first as organizers of information to model, represent a scenario or situation. So, we have a situation like the pack and sticks of gum that we could literally say, "Hey, let's sort of organize this information. We can kind of put it in this table, and it would represent that pack to those sticks and different numbers of packs to sticks that always had, always represented 1 pack to 27 sticks. In other words, anywhere in that table, every one of those packs had 27 sticks in it. That makes it a ratio table, and we can sort of organize information. That's kind of the first way. 

But secondly, we can also use ratio tables as tools to represent strategies for multiplication and division. So, as we develop alternative strategies with students, and students are using relationships and connections to multiply and divide, we, as teachers, can come in and represent their thinking using ratio tables. Now, we can also represent their thinking using open arrays, the area model. We can also use equations. But one of the tools that is so powerful to represent their strategies, those relationships they're using for multiplication and division, is a ratio table. That is another way that we can represent the way they're thinking about solving multiplication and division problems. 

Third, ratio tables can be used as actual tools for solving multiplication division problems. Like, they actually become the way that I begin to think multiplicatively to solve multiplication division problems. They're actually tools to solve. So, let me just say that again. One, they're kind of organizers of information, they sort of model the situation, they represent what's happening. And then we want to kind of move students. We want to help them transition to modeling their strategies, representing what they do, the relationships they use to solve problems. And then lastly, we want to transition students to actually use the ratio table as a tool to help them keep track of the relationships they're using. And they actually use it to help them solve multiplication and division problems. Now, we then could go to middle school and continue to have that go, and then use that ratio table as the proportional tool that it is to solve proportions using proportional reasoning. But today we kind of want to talk about how ratio tables could be used in these multiplication and division situations, these multiplicative reasoning situations. Because as students begin to use ratio tables, they learn to scale in tandem. As I double the packs of gum, I double the sticks of gum that I have. As I multiply the packs of gum times 10, I multiply the number of sticks of gum times 10. That I'm sort of scaling in tandem. And that act of scaling in tandem is leading toward this thing that they'll do when they are solving proportions with non-unit ratesAbsolutely. And it's not really the facts themselves that are the issue. It's the pressure surrounding them and the ways of practicing that people argue over. So, let's spend a minute defining and clarifying.

Yeah, so let's do. Alright, so what do we mean when we say multiplication facts or multiplication tables? What does that mean? I don't actually use multiplication tables very often, but the more I work internationally, the more I hear people talking about multiplication tables. So, we want students to know single-digit multiplication facts. Sometimes some standards call for multiplication facts through 10 by 10, sometimes 12 by 12. So, that's like sometimes... I even just heard somebody say that their kids have to memorize through their 15s.

Kim  18:51  
Right.

Pam  18:51  
Ah, that's like maybe a little bit a lot. So, for example, when I say 10 by 10 or 12 by 12 or through the 15s if we were to pick like 7s. So, 7 times 1, 7 times 2, 7 times 3, all the way to if we're doing 10 by 10, all the way to seven times 10, if we're doing the 12s, all the way through 7 times 12. Or even for that one teacher, 7 times 15. We're having kids sort of know those multiplication facts, multiplication tables.

Kim  19:17  
Yeah. And most of the time that someone talks about those facts, they hear recall from rote memory. And, Pam, sometimes you get misheard. People have mistakenly thought in the past that you think that kids don't need to know their facts. Can you tell us more about what you really mean here? Because here in the States, the standards say students need to know their facts. In fact, this Common Core standards in grade three. I'm going to quote this. It says, "Fluently multiply and divide within 100, using strategies such as the relationship between multiplication, division..." And they gives an example. "...or properties of operations." And it says, "By the end of third grade, know from memory all products of two one-digit numbers." And here in the State of Texas, where we are, our standards say, "Recall facts to multiply up to 10 by 10 with automaticity."

Pam  20:08  
Yeah. So, this gets a little sticky, so let's be clear. Does Pam Harris think students need to know their facts? Yes. But let's define "know". What does it mean to know their facts? And I used the word "know" on purpose. What I didn't say was, "Does Pam Harris think students need to memorize their facts?" because the word memorize is tricky because it means different things to different people. It's a tricky word, and I think that when we use the word "memorize", and even when we use the word "know" and some other things, that we kind of talk past each other because we assume the other person knows what we mean. So, let's define knowing versus rote memorizing.

Kim  20:46  
Okay.

Pam  20:46  
Because the wording of these standards is problematic. They try to walk the line. They try to please everyone in the way that they word it. And because of that, it gets muddy. So, let's clear it up. What's the difference between rote memorizing and knowing? So, I don't even use "memorizing" anymore because it's so, so confusing, and so whatever. So, rote memorizing and knowing. So, defining some terms. I'm going to define "rote memorizing" is recall without meaning. And "knowing" is to have at your fingertips. I believe that that's what the standards mean when they say "automaticity" and when they say "know all products of two one-digit numbers," they mean know, to have at your fingertips to be able to use, to be able to not haltingly spend forever on or not know at all. Like if a student is, "I don't know," then obviously they're not being able to pull. Or but it also means the student who it doesn't take five minutes to figure out the fact.

Kim  21:48  
Right.

Pam  21:49  
Either one of those. To "know" means that we sort of have it at your fingertips. It's in your psyche. But rote memorizing to recall without meaning is not, what I would suggest, is not the intent of the standards.

Kim  22:03  
And so, you said "automaticity", so let's define that. What does "automaticity" mean? We believe it means what people mean when they say, "know your facts". They mean have them at your fingertips, be able to use them, and really they don't have to fuss or struggle or reach without success.

Pam  22:21  
Yeah, and I believe that math people came up with the term "automaticity" and "fluency" because they see some students fluently using facts and other students not. And they wanted to be able to help us all think about the difference between rote memorizing, that recall without meaning, and knowing, which is owning down deep. Like, you just said, that automaticity.

Kim  22:43  
Yeah, I think it's a great time for you to share the quote that you wanted to talk about with Cathy Fosnot in her Young Mathematicians at Work series.

Pam  22:50  
So, she has a section in her books called Memorization or Automaticity? And she says, "Memorization of basic facts usually refers to committing the results of operations to memory, so that thinking is unnecessary. Isolated multiplications and divisions are practiced one after another. The emphasis is on recalling the answers. Teaching facts for automaticity, in contrast, relies on thinking. Answers to facts must be automatic, produced in only a few seconds. Counting isn't sufficient. But thinking about the relationships among the facts is critical. A child who thinks of 9 times 6 as 10 times 6 minus one 6 produces the answer of 54 quickly. But thinking, not memorizing..." and I'm going to put the word in here, rote memorizing, "...is at the core. Although, over time these facts are remembered. The issue here is not whether facts should eventually be memorized..." I would put in parentheses "known", "...but how this memorization, [knowing]..." That's my word, "knowing", "...is achieved by rote drill and practice or by focusing on relationships," unquote. And then she talks about ways to develop relationships and says, "In this way, the facts become automatic. But the relationships, the heart of mathematics, are not sacrificed." That's brilliant, right? The relationship, which is the heart of mathematics, are not sacrificed.

Kim  24:25  
Grades 3-5 is where we move students from additive to multiplicative thinking and spend a lot of time with multiplication and division using really important models that extend even beyond these grades. We love grades 3-5 teachers, and we're so happy that you are helping to make math figure-out-able for your students. Keep up the great work.