[00:00:01.470] - Pam
Hey, fellow math-ers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned math-er.

[00:00:11.030] - Kim
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.

[00:00:19.010] - Pam
We know that algorithms are super cool human 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.

[00:00:32.540] - Kim
In this podcast, we help you teach math-ing, building relationships with your students, and grappling with mathematical relationships.

[00:00:39.210] - Pam
Y'all, we're so glad you're joining us to make math more Figure-Out-Able.

[00:00:44.400] - Kim
How many times a day do you think you say figure-out-able?

[00:00:48.500] - Pam
Golly. On podcast recording day, about 100. Non-podcast recording days? At least a few times.

[00:00:57.250] - Kim
Yeah.

[00:00:57.640] - Pam
Somebody contacted me the other day and said that I should... Oh, should I even go there?

[00:01:02.710] - Kim
Probably not. Go ahead.

[00:01:05.590] - Pam
That I should sue the gal who has Everything is FigureOutAble. They said, hey, I saw this thing in Target that was Everything is FigureOutAble. Maybe I'll just tell everybody. We are on fine terms with... I don't remember her name, but there's a self-help person out there who trademarked Everything is FigureOutAble. We trademarked Math is FigureOutAble. We have agreed to let each other. You know, you're pretty much not going to go into a bookstore and buy a self-help book thinking you're buying a math teaching book. And you're not going to buy a Math is FigureOutAble teaching book thinking that you're buying a self-help book. Like, it's just, you're not. And so, we kind of agreed that we could both use it, and it's good. So, yeah. So, we're all happy. You don't need to get mad at anybody that uses Everything is FigureOutAble. We're all good. We had a nice conversation. It's all good. Yeah.

[00:01:47.960] - Kim
Fabulous.

[00:01:50.020] - Pam
We like to get along. It's good. Yeah.

[00:01:52.860] - Kim
It's funny that you mentioned books because what I was going to say when we started this episode was that one of the, I think, really unique things is that you spend a ton of time talking about K-12 stuff. And really, like for adults too, and leaders, and whatever. But very vertical, right? Like, Math is Figure-Out-Able has put out books at every grade level, multiple at every grade level. And I think that's kind of unique. I think it's interesting that you have such a wide perspective on mathematics. And, you know, we're going to talk today about being vertical. (unclear).

[00:02:30.310] - Pam
Have you heard this story before? I'm going to tell it really fast.

[00:02:32.720] - Kim
Okay.

[00:02:33.430] - Pam
When I was totally secondary, and I was in the T3™ community. So, Teachers Teaching with Technology. It's the TI group of teachers that are expert facilitators teaching people how to teach with graphing calculators. And I got in really young because I was... For lots of reasons. Anyway, so I was very active. I was teaching teachers how to teach with graphing calculators, super technology-oriented. And I went to one of their trainings, and Janie Schielack, Dr. Janie Schielack out of A&M. She's now retired. I think Vince is still going strong. Somebody told me the other day at A&M, Texas A&M. And one day I asked her. I was like, "Hey, I'm kind of interested to learn more about like younger math." And I'll never forget, she just lit up. Like, her face was just like. It's like I'd just given her chocolate cake or something. She just was like, "Ah!" She goes, "Pam, go there." And I was like, like, "What?" She's like, "Oh, go K-12.  It's such a small niche, and so few people really are good K-12. Like, people just don't even attempt it so much that you will thrive there." And man, I'm so glad she gave me that advice.

[00:03:34.930] - Pam
It was such... You know, I was really young. It was such good advice. Really impacted my career. I love the verticality of it.

[00:03:41.100] - Kim
Yeah. And I think, you know, there's a lot of areas where people recognize it's important to focus K-12. Like, let me give you an example. So, in our standards, in our state standards, there are the process or practice standards that are vertical K-12. So, these are the mathematical behaviors, what we want kids to be able to do, and the ways that we want them to behave, and justify their thinking. And all of those standards are the same K-12 because those are important. Then in the content standards, it's clearly one grade leads from the one before, and so we can follow specific like addition, or multiplication, or whatever. And they lead from one to the next. It's, you know, it doesn't make these giant jumps. Vocabulary is another one where people talk about the need for verticality. You know, as a 3rd grade or 4th grade teacher, I would hear from middle school teachers, "Don't say it this way. We need it to be consistent across the grades." And so, you know, that's an area where we talk about being K-12. Notation is another one, where, you know, we really want grade to grade to grade to have as much stability as you possibly can.

[00:04:52.870] - Pam
Yep.

[00:04:53.350] - Kim
One of the things that we talk a ton about here at Math is Figure-Out-Able is major strategies for operations. That we don't want to name something one way, and then name it something different. And we don't want the actions that are happening. You know, we've got some strategies that can be used, and we name them a certain way. That they are applicable in K-1, 2, 3, 4, 5, and on. Especially some of the major, most sophisticated strategies. We love the Over strategies and the equivalent strategies because they are in many operations, and they flow K-12. So, it's not about like something new all the time.

[00:05:33.050] - Pam
And many different, I don't know, sizes of numbers is the right word.

[00:05:36.160] - Kim
Yeah.

[00:05:36.510] - Pam
Trickiest. Like, I can start I can Over with single-digit, Over with double, triple, into the thousands and millions, but I can also Over with fractions and with decimals. You can Over with percents. And like, there's lots of Over sort of is very vertical, right? It's one of those strategies we emphasize because it is so vertical. And the equivalent strategies like Give and Take or Constant Difference or Flexible Factoring, or thinking about division, quotatively and partitively, those all grow up. And those equivalent strategies lead to equation solving and understanding with equation solving, not just mimicking a bunch of steps to solve equations, but really feeling the equivalence in an equation. The equal sign isn't just to do it, it actually represents equality. Yeah, nice.

[00:06:20.880] - Kim
Yeah, so I think if you ask somebody, "Is it important to think vertically from K-12?" I think the answer people would say is, "Yes, of course." And we talk about these particular areas that I just mentioned. But what about models? We had mentioned Lorna last week actually. And Lorna had sent in this question. She said, "I'm sure this will be in the 6-8 and 9-12 books." She's mentioning the grade level companion books for Developing Mathematical Reasoning.

[00:06:49.150] - Pam
Yeah, nice. Thanks, Lorna.

[00:06:51.170] - Kim
Which are fantastic. And she says...

[00:06:53.760] - Pam
Maybe I'll just say. Sorry. So, Developing Mathematical Reasoning, K-12 came out in 2025. Then the accompanying K-2 book came out in 2025. '26, the 3-5 book, grade bands 3-5 came out. 6-8 comes out September of 2026. And the 9-12 book comes out February of 2027. So, that's 5 books in 3 years. Everybody take a deep breath for me. It's been a big project. But we're super happy about it. Okay, so Lorna said.

[00:07:20.490] - Kim
She said, "I'm sure this will be in the 6-8 and 9-12 books, but something on..." She'd like the podcast please. "...on something how the major strategies and/or the models grow up as we progress through mathematical reasoning. For example, I harp on about the area model for multiplication and division as it grows up to polynomial multiplication/division. But it needs to be taught as true area with proportionality to begin. Otherwise, it becomes another algorithm rather than a way to reason."

[00:07:48.850] - Pam
Nicely said.

[00:07:49.410] - Kim
So, here you go, Lorna.

[00:07:50.860] - Pam
Yeah, alright. So, I'm going to start with Lorna's model and agree with her that if in early multiplication, we actually use a true area model. Now, podcast listeners, you might be like, "Well, I know an area model. That's where you draw a box, and you split it into 4 equal parts." And as soon as you start thinking that, I invite you to consider that's not an area model because what you just did was draw a square to represent every multiplication problem. Most of which are not square. Like, 7 times 8 is kind of square. It's mostly square, but it's not square. And maybe I could even do 15 times 18 is not square. It's squarish, right? It's more square than 2 times 18. But 15 by 18, I wouldn't want to represent it in a square, and then always cut the 18 into 10 and 8 and the 15 into 10 and 5, and then multiply the tens and multiply all the pieces together and add them up. We can do a little bit of that as kids are sort of making sense of double-digit multiplication. Maybe even single by double-digit multiplication. But then quickly we want to ask them...
[00:08:55.300] 
Well, first of all, we want to keep those area models proportional, like Lorna said, so we want that 15 by 18 to look less deep. 15 is the number of rows and 18 is the number of columns, so it should be less deep than it is wide. Just by a little bit. Just by 3, you know, because 15, 18. And then we want kids to think, do we have to cut that into 10 and 8 and 10 and 5? No way. We could do 10 times 18. We think about 15 times 18 as 10 by 18. That's a 10 by 18 rectangle. And then we can add onto that a 5 by 18. Well, we know 10 by 18 is 180. 5 by 18, that's just half of it. So, bam, we don't have to know 5 times 18. We can just halve the 10 times 18. And the 10 times 18 is 180. Half the 180, we get 90. Add those together, and now you have 15 times 18 is 270. If we do that young, A, we could represent A, kids are getting better at area. They're chunking numbers in ways that make sense.
[00:09:55.280]
We can use it to really feel and understand doubling and halving strategies and flexible factoring strategies that grow up into decimal multiplication. We can represent decimal multiplication. Not for very long because it gets very hard to be proportional. But we do enough of it that kids are like, "Oh, yeah. I can really feel this tenth by 4. You know, if you're doing something like 2 and a tenth by 4, then you're going to have a 2 by 4 and you're going to have a tenth by 4. And I can think about what is the area of a rectangle that is 1 tenth of a unit by 4 units. And then I could do the same and I was just actually picturing decimals when I said that. I guess I could have said 4.1 because you do the same thing with fractions, 4 and 1/10, 2 and 1/10 times 4. And so, that's a nice model, not only for students to actually understand area as a 2-dimensional measurement system, but to also then be able to say, "What does it mean when I don't know one of the dimensions, but we want to sort of generally represent what's happening with multiplication?
[00:10:52.970]
Can I think about x times 18? And can I think about x plus 5, the quantity x plus 5, times 18? Does that mean I've got x times 18? And 5 times 18?" I'm sorry, I'm trying to connect that to our 10 by 18 and our 5 by 18. If I've got x plus 5, quantity, times 18, it's kind of like I've got a rectangle where I don't know what this dimension was. So, that x by 18 plus another rectangle, that's the 5 by 18. And bam, all of a sudden I have... What did I just say? x plus 5. Now, I have to start writing. I've been all in my head. x plus 5, all times 18 can become that x by 18 plus that 5 by 18. And we've just done the distributive property with variables. That is a desirable outcome. Kids are actually understanding what's going on, drawing back on their real understanding of an area model. Now, to be clear, once we involve x's, we're no longer talking area because x could be negative, and a negative dimension doesn't really make sense unless we get all Einsteinian and we're sort of like in black hole area or antimatter or something.
[00:11:55.980] 
But we can absolutely let that model grow up, like Lorna said, to polynomial multiplication and maybe even more importantly polynomial division. We do not need the long division algorithm. We can use what we are going to call an open array model to help kids reason through polynomial multiplication/division. So, that's one model that grows up. Thank you, Lorna, for that one. There are other models that are super important because they grow up through the grades. Another one is a number line, and maybe specifically an open number line. So, an open number line is the one where you're only putting tick marks where you want to. And sometimes, we've kind of given elementary teachers, I don't know, some grief a little bit. Like, "Why are you making kids draw these pictures?" When in reality, an open number line is not a picture. It's not a crutch for kids who can't do it the right way. It's a measurement model. Like, that open number line is based on a ruler. Which means I'm really thinking about the span between numbers. That's going to be super important for kids to realize. So, we're not only using an open number line to add and subtract, but we're building measurement, linear measurement.
[00:13:04.820] 
Just like we were building area, two-dimensional measurement, when we use an area model, now with a linear model, we're building one-dimensional measurement. Well, that grows up into a double open number line. In the middle school, we want to use proportional relationships. And, y'all, that grows up. Now, put in a horizontal number line perpendicular to a vertical number line. And bam, we have a coordinate axis where Descartes, the guy who said, "I think, therefore I am," is also the guy who said... He looked up from his bed, saw the fly on the ceiling, and he said, "Huh, how could I tell somebody where that fly is?" Well, I think if I define where the origin is here, two perpendicular lines, and I defined where they meet as the origin, I could talk about how far over it is and how far up it is from there. And I could sort of use this ordered pair to talk about where that point is." That's an extension. That's a growing up of the open number line. So, thanks, Lorna, for asking us about models that grow up. I've now talked about area models, talked about number lines. One other model that really, Kim, you know, we love that maybe not everybody has learned to love as much as we do, yet.
[00:14:15.570] 
We're getting it going. We're getting it going around the world. Is the ratio table. We've just recently met with some wonderful people who are really thinking hard about mathematics education, and they don't use ratio tables in grades 3, 4, and 5. But boy, they're considering it. And we want to keep working with them. Ooh, go back to the Results Pyramid. We want to give them more experiences that can help influence their beliefs about ratio tables. So, a ratio table is a specific or special paired number table. So, we've paired numbers. They're on a table. It's a ratio table when the ratios of those pairs are equivalent. So, if I had something like a pack of gum had 9... I'll do 17. 17 sticks in that pack, then 2 packs would have 34 sticks and 4 packs would have 68 sticks. 1 to 17, 2 to 34, 4 to 68. Those are all equivalent ratios. So, if the entries in the table are proportional, equivalent ratios, then we call that a ratio table. We like to use ratio tables as young as 3rd grade, right, Kim?

[00:15:21.300] - Kim
Yeah, absolutely.

[00:15:22.450] - Pam
So, 3rd grade, we've got kids talking about... Golly, bags. Help me. What has 3 in it, Kim? Something has 3 in it.

[00:15:30.940] - Kim
Table groups. 3 kids in a table group.

[00:15:32.700] - Pam
Oh, nice. 3 kids in a table group. Ooh, I just thought of 3 tennis balls in a thing, in a container. Let's do the table groups. So, if I've got 1 table group, I've got 3 kids. If I have 2 table groups, I've got 6 kids. If I've got 3 table groups, I've got 9 kids. But we don't necessarily have to go in order or all the same. At that point, I might say, "Well, you got 1 table group, 2 table groups, 3 table groups. How about 5 table groups?"

[00:15:52.790] - Kim
Mmhm.

[00:15:53.330] - Pam
Bam. Like they could say, "Well, 2 table groups is 6 kids, 3 table groups is 9 kids. So, if I add the 2 and the 3 table groups together to get 5 table groups, I can add the 6 and the 9 students together to get 15." And it's a beautiful way of getting kids to start thinking in terms of bigger chunks of numbers. Just like we do with the area model, but it's a different model that is also building sort of young or less sophisticated proportional reasoning at the same time. We... Oh, go ahead.

[00:16:20.030] - Kim
Yeah, I was going to say one of the really important things about ratio tables younger that I think we would say is when you just mentioned using an area model, and you talked about once we start multiplying with decimals, if we buy the story that area models really represent area, and we in 4th grade start working with decimals, 4th and 5th grade multiplying decimals, then we need the ratio table at those ages, so that we can start multiplying bigger numbers, smaller numbers, and maintain proportionality. Because we're not going to represent 23.6 times 15.5 well on an area model. So, then at that point we would switch to ratio table.

[00:17:01.630] - Pam
Especially if it's 235.5.

[00:17:05.750] - Kim
Yeah, even better.

[00:17:05.870] - Pam
Times 2.1.

[00:17:07.050] - Kim
Yeah.

[00:17:07.630] - Pam
There's no way you're going to get that 0.1 and that 200 on the same board, right?

[00:17:12.120] - Kim
We shouldn't have to sacrifice the proportional thinking in order to mess with larger and smaller numbers.

[00:17:17.890] - Pam
That is really nicely said. I like that. Nice job. So, we could do that with single-digits like I just did with the 3 kids at a table group, but we can do that with much bigger numbers like I did earlier with the 1 pack to 17 sticks. We can do that with crazy numbers. Like, I don't know. Give me what was the decimal you just said 4.3 or something?

[00:17:35.610] - Kim
23.6.

[00:17:36.740] - Pam
Okay, or that.

[00:17:37.550] - Kim
Whatever. Pick a number.

[00:17:39.380] - Pam
Yeah, so 23.6. So, I could say, hey, if one, golly, bottle has 23.6 ounces in it, then how many would 2 bottles have, or 10 bottles, or 5 bottles? And bam, kids could really be reasoning about... In fact, should we even do that? 236. So, 5 bottles would be 118 ounces. And then I could say about 15 bottles and 16 bottles. But I could also say half of a bottle and half... Oh, golly. Half of a bottle would be 11.8 ounces. And you know what? I have to tell you what I just did was I was going to cut the 23.6 in half, and I thought, don't do that. Just scale from the 5. It was much easier because I already had the 5 at 118. So, now we can do 5.5. And I could do how about a tenth of a bottle? That's kind of a funny thing to think about, but that would be what? 2.36 ounces. And so, now I could do 5.6... Bottles is funny. But 5.6 times 23.6. And bam, I've just reasoned through that problem maintaining proportionality because I could use that ratio table.

[00:18:43.150] - Kim
Yeah.

[00:18:44.030] - Pam
That ratio table is going to continue to grow up to become a model that we're going to use to represent proportional relations. And so, now we're going to graph that sucker, and we're going to look at the equation of a line that goes through the origin. That line's then going to get shifted and become a non-proportional relation. And now, we're out of a ratio table, but it built the background, the basis for linear functions. And linear functions are the parent function of all functions. So, lots of nice verticality with ratio tables. We really hope that teachers and curriculum writers accept the challenge of bringing ratio tables in earlier than they might have. As soon as kids start thinking about groups of things, we can bring ratio tables in. Third grade for sure. All the way up through proportional middle school stuff into writing the equation of a line in sort of high school classes.

[00:19:31.790] - Kim
Yeah. And it's not necessarily about forcing kids to draw these models for everything for every time. That's not what we're suggesting. But we are suggesting these are models that are super important because we know that as we increase the size of numbers, or we shift to decimals, or shift to fractions, or we start doing stuff in middle school, they have that as an anchor. They have models as an anchor to connect to what they already know. And man, if we can help students see a thread throughout their education, we're doing them such a service.

[00:20:02.410] - Pam
And not only see a thread, but feel that thread and use that thread. And with every one of these, we're not just getting answers to questions, but we're also building mental maps in our heads.

[00:20:13.900] - Kim
Yeah, that's nice.

[00:20:14.890] - Pam
And nice connections and dense connections. Lots of relationships that we can use. And these particular models that we mentioned today are super important in part because they are so long-lasting. So, like you said, it's not about picture drawing. It's not fuzzy, dumbing it down. It's about using models to help brains grow. Any last words, Kim?

[00:20:36.880] - Kim
Hmm.

[00:20:37.450] - Pam
Go vertical models. Go, Lorna. Thanks for the question. Y'all, if you want to ask us a question that you want us to answer on the podcast, give us a good review or join our teacher or leader coaching groups, and we'll get one of those answered here. Y'all, thanks for joining in teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!