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

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  0:19  
We know that algorithms are super cool human achievements, but y'all, they're 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  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:39  
We invite you to join us to make math more figure-out-able. Hey, Kim.

Kim  0:44  
Hi.

Pam  0:45  
Howdy, howdy, howdy. Okay, so it's warming up a little bit. It's been crazy cold this year. 

Kim  0:49  
Yeah, super cold. 

Pam  0:50  
Yay.

Kim  0:51  
Can I tell you something I just thought about? It's like a podcast thing.

Pam  0:55  
Okay.

Kim  0:56  
So, when you're talking is when I like will make a make a noise or move something because...

Pam  1:03  
Totally, yeah.

Kim  1:04  
I just heard you moving while I was talking. 

Pam  1:07  
While you're talking. I waited. I waited until you were talking. I was like, Ooh, shift my chair. I'm not touching the microphone. Yep, yep. 

Kim  1:13  
That's exactly what I thought you were doing because that gets edited out.

Pam  1:18  
I actually also moved. So, when I do a virtual presentation, I have my cards. I have those 5 by 7 cards.

Kim  1:23  
Yeah, yeah.

Pam  1:24  
Or 4 by 6, 5 by... I don't know what size they are. Anyway, they're cards. And they sit on my microphone. Like, on the face.

Kim  1:30  
Oh, yeah.

Pam  1:31  
Which means... Anyway, so I moved the microphone closer. So, while you were talking, I moved the cards. I moved the mic closer, then I shifted my chair. 

Kim  1:38  
I knew it. I knew it

Pam  1:39  
You heard the chair.

Kim  1:40  
I hear the chair.

Pam  1:41  
So, now Craig's like, "Does that mean I cut it out? Do I not cut it out?"

Kim  1:44  
Oh, that would be funny. Yeah.

Pam  1:45  
Craig, I'll let you decide whether you...

Kim  1:47  
We have told Craig that we need bloopers, right? Oh, gosh. 

Pam  1:50  
Oh, that's true. Yeah.

Kim  1:51  
That would be a mess.

Pam  1:53  
I think the bloopers are just a mess. Yeah. That's the definition of a blooper. 

Kim  1:57  
Yeah. 

Pam  1:57  
Hey, Kim. So, I had this really cool experience yesterday.

Kim  2:01  
Oh, okay.

Pam  2:01  
Well, actually, last fall, I had a really cool experience that I was invited to present to ASSM the Association of State Supervisors of Mathematics.

Kim  2:09  
Yeah.

Pam  2:09  
Pretty prestigious group to belong to. 

Kim  2:11  
Yeah, yeah.

Pam  2:11  
And so, that was wonderful. They invited me back to do a virtual, sort of follow-up session, and I did that yesterday. It was really cool. And one of the participants, who I'm a little embarrassed to say I don't remember her name, so sorry. But she told a story about how she and her daughter will listen to our podcast episodes in the car, and that her daughter, after a couple of them, said something about... It was a MathStratChat episode, and the daughter was solving the problems. And she got in the car the next day, and she says, "Are you listening to that figure-out-able podcast again?" So, this is a shout out to Miriam for listening to the Math is Figure-Out-Able Podcast with your mom. Thanks for listening, and thanks for solving the problems with your mom, making math figure-out-able for yourself. Way to go. 

Kim  2:57  
That's so fun.

Pam  2:58  
Said I give her a shout out. Mom gave me permission to use her name. So, there you go.

Kim  3:01  
That's very cool. I love how you do the kid's name, but not the mom. 

Pam  3:05  
Yeah.

Kim  3:06  
That's amazing.

Pam  3:07  
I'm pretty sure it's because I was supposed to write down the mom's name. I mean, I could see it on Zoom. And I'm lame and didn't. But I did write down Miriam.

Kim  3:13  
Well, way to go, Miriam. Okay, let's get into today. It is a really, really exciting time here at Math is Figure-Out-Able. So many fantastic things. A year ago, Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, the K-12 anchor book came out. That book was an amazing release. It's talking about how and what, big ideas about developing mathematical reasoning. And then last September, September 2025, the K-2 book, came out. The what-to-go-now-do-in-your-classroom K-2 book. And now it is time. The one I've been most excited about, the 3-5 version is here. It's time. It is here. Yeah, it's... You know, I think listeners know that I spent my teaching time in 3-5. 3, 4, 5. And I got to tell you. You know, obviously, like I've seen the book.

Pam  4:15  
You read the book. You helped edit the book. 

Kim  4:17  
I think this is such a gift to third, fourth, and fifth grade teachers. You know, it's the anchor book, the one we just talked about, the K-12 book is so powerful for shifting your thinking. And, you know, now getting the 3-5 book, it gives so many supports and so much help to what to then go do. 

Pam  4:39  
Grades 3-5, yeah. Thank you. And coming from you, that is high praise because that is the land where you lived.

Kim  4:46  
Yeah.

Pam  4:46  
Yeah, I really appreciate that. So, the title everybody. Developing Mathematical Reasoning: The Strategies, Models and Lessons to Teach the Big Ideas in 3-5. And it is out! So, you've been able to order it, pre-order it for a hot minute, but it should be shipping when this podcast drops. Fingers crossed. Should be.

Kim  5:05  
Yeah.

Pam  5:05  
Yeah, so we're super excited. It's been a lot of good work. If you have not yet looked at the K-12 anchor book that Kim just mentioned that came out a year ago, that is the book that many school districts across the country are using as their new book study. Definitely check that out. Is a uniquely K-12 book discussing math and math teaching from kindergarten through twelfth grade. So, make sure you check out that anchor book. But, Kim, today, let's focus on the 3-5 focused book about developing mathematical reasoning. 

Kim  5:37  
Yeah, yeah. So, before I buy a book. You know, I go to the bookstore or get online. The first thing I do is flip to the beginning to say what is in this book, what are the chapters? So, let's dig in to what... 

Pam  5:52  
Do you read back? I'm curious.

Kim  5:54  
What do you mean? 

Pam  5:55  
The back of the book? Like, do you read the, you know.

Kim  5:56  
The summary on the back cover?

Pam  5:59  
I don't know. Yeah, like whatever's on the back cover. Do you? 

Kim  6:00  
Oh. Yeah, sometimes.

Pam  6:01  
Do you read the acknowledgements? 

Kim  6:04  
I mean, eventually, if I buy it. But otherwise, no.

Pam  6:07  
No? Oh, okay, but eventually. So, like, if you read a book, do you read all? I saw a post the other day on social media where the author was like, "Who reads every single word in a book and who ignores all of the fine print?"

Kim  6:19  
Yeah, I mean if it's somebody that I know that writes it, then I'll read the acknowledgements. But regularly, not really. 

Pam  6:26  
Just a random, you're like, you don't care who they...

Kim  6:28  
Yeah, yeah.

Pam  6:29  
Yeah, sure.

Kim  6:31  
But I do look at the table of contents because I want to get a feel for what's in the book. So, this book has four parts, and we're going to dig into what the parts are and what they're about. But I have a question for you before we even get into that. 

Pam  6:45  
Okay.

Kim  6:46  
Can you tell me what is one favorite thing of yours in this book? 

Pam  6:51  
Mmm, yes. So, I have to choose. But one thing that stands out when you just ask that right now is that we spent, we took a lot of effort and care in putting what we call Problem String walkthroughs. So, we take an actual video of a Problem String that we've done with real kids, and that we have it on video, and we have the transcript, and we help you... What are you laughing about? Oh, real kids.

Kim  7:17  
When you say real kids.

Pam  7:20  
Not fake kids.

Kim  7:21  
Not fake kids.

Pam  7:23  
So, we take that, and we write it up in such a way to help teachers really get a feel for what the kind of math-ing that we're advocating actually looks like, and feels like, and sounds like. And you get the perspective of the experience that so many of us don't have when we were kids. So, like, you know.

Kim  7:47  
Yeah.

Pam  7:47  
When we were growing up in math classes, most of us had a very similar kind of same experience of what it meant to rote memorize and mimic, and so how can we know to do anything differently? Well, the best we can in a book, we did these. I don't know. There's got to be a better word. But sort of story lines where we really help you get a feel for the experience of what it means to really teach math-ing and develop students as math-ers where they are doing the mental actions of math-ing. 

Kim  8:19  
Yeah. So...

Pam  8:20  
Really happy about that. 

Kim  8:21  
So, you kind of unfold the Problem String in the classroom as it's happening. It's got some images. And it's got like what the teachers and students are saying, and there's some commentary about the teacher moves. Like, calling that out. So, yeah.

Pam  8:33  
Exactly that, yeah. Big, beautiful pictures of these great kids.

Kim  8:36  
Yeah.

Pam  8:37  
And then for many of them, there's a QR code that you can actually then go watch the video. 

Kim  8:42  
Yeah.

Pam  8:43  
When you're reading, you can use the QR code to go actually watch the real video. So, yeah.

Kim  8:46  
Yeah, yeah.

Pam  8:47  
Really, that is a favorite thing in the book. 

Kim  8:49  
Okay. Well, it felt like maybe you were going to say one more. No? Because you said when I asked you right now.

Pam  8:55  
Yeah, if I have to choose.

Kim  8:56  
Do you have a second?

Pam  8:57  
If I have to choose another one, I will say that we often get asked questions about types of tasks and about models.

Kim  9:05  
Mmhm.

Pam  9:05  
And we've done a lot about that on the podcast and in different places. But this is one place where if you teach grades three, four, and five, where you can really get probably the most in depth all together in one place, sort of treatment about the types of tasks that we really like that work really well. And then also models and modeling. And what does it really mean to help students create mental maps in their heads like mathematicians have, to use those to math. So, I think that's another favorite. Yeah. 

Kim  9:37  
Okay.

Pam  9:38  
By the way, everything that I just said is also true of the K-2 book.

Kim  9:41  
Yeah. Yeah, yeah.

Pam  9:43  
Yeah, yeah. Cool.

Kim  9:44  
Alright, so this book has four parts.

Pam  9:47  
Mmhm. 

Kim  9:47  
So, I'm just going to... Let's just go through the parts. The first part is called Setting the Stage. What's in that particular section? 

Pam  9:54  
Yes, thank you. So, there's a preface and chapter one in that part that are really about developing mathematical reasoning. So, again, you see the graphic that we talk about all the time that goes from counting strategies to additive thinking to multiplicative reasoning to proportional reasoning to functional reasoning. Those hierarchical stages. It helps lay that groundwork again. That's in the K-12 book, we really dive into much more detail about all of those domains.

Kim  10:20  
Yeah.

Pam  10:21  
But for K-2 we re-talk about that, but we also introduce the shark metaphor. So, we've talked about the shark metaphor before on the podcast. Right? Haven't we? I'm sure we have.

Kim  10:32  
Yeah. Yeah, yeah.

Pam  10:34  
Never know where I've done what. But now it's in the book, and so you can actually see the autostereogram and see and hear that metaphor. If you haven't ever heard about that, we'll put in the show notes the episode where we talk about the shark metaphor and what it means to math. It's a really helpful way, we found, to really have a conversation about the mental actions that you used as a student versus the mental actions that we now know we can help students develop and how we can do that. So, it really sets the stage, which is the name of the part. Sets the stage for what to do to help students learn to math. 

Kim  11:07  
Yeah. I love that that part is in that, that analogy, is in the book because when you are doing a book study with your team or with your school, you can really like read that part together, and then take the time to unpack what it means to you in a way that if you, you know, see you do that live, that, you know, you have this movement in your mind. You're like, "Okay, I totally get what you're saying," but then it's gone. You can't revisit it, and you can't kind of unpack it yourself maybe to your teammates. So, I was really glad that you wanted to put that in the book. Part two, Developing Additive Reasoning. 

Pam  11:45  
Yeah, so consider grades 3-5. What are the domains that we really are developing in 3-5? And the first one is additive reasoning. Even though we start developing additive reasoning in grades one and two with, you know, smaller numbers, less complicated numbers, in three, four, and five, we really continue to develop additive reasoning. Some teachers might be like, "Uh, I still got kids counting by ones." I know. And so, we need to, you know, give you help at getting kids out of counting strategies into additive thinking. And then with crazier, more complicated numbers. So, bigger numbers. We deal with higher place values. But also decimals and some fractions. You know, helping... Some fractions... And fractions. Both with the major addition strategies and also with the major subtraction strategies. So, we dive into what are the major relationships that kids need to develop, so that the major addition and the major subtraction strategies become natural outcomes. The things that occur to students when they see crazy numbers. And how to develop those. So, Problem Strings to help students really develop those major addition and subtraction strategies. 

Kim  12:53  
Yeah. Sample Problem Strings that people can use in their classrooms.

Pam  12:57  
Yeah. And speaking of QR codes, there's also QR codes throughout the book where you will, like we said before, you'll kind of see a Problem String in action. And then it will say, "For this major strategy, you can download kind of a cheat sheet to go try this Problem String with your students." And we've given those throughout the book for all of the major strategies to help teachers really have, like you said, the things to go do. What can I now go try with my students and find success? 

Kim  13:25  
Yeah. I don't know if you're going to mention this, so I'm just going to mention it now that one of the things I love kind of all throughout the book is that there are some call outs for like tips. So...

Pam  13:35  
Mmm, mmhm.

Kim  13:35  
...I don't think we've said that yet. But when you're reading along, you know, you might have a question in your head, and then you scan down, and it's like you already know the things that people are going to ask, the tip, or the "Hey, heads up."

Pam  13:49  
Yeah, we call some of those Frequently Asked Questions sections. 

Kim  13:52  
Yeah

Pam  13:52  
And I think that's exactly. People have said they'll be reading, and then all of a sudden, they're like, "But what about..." And then the next like,  "Frequently asked question. What about this?" And then we... Yeah, we have some answers.

Kim  14:01  
Yeah, yeah. 

Pam  14:03  
I'm glad. Yes, I'm glad that we were able to put those in all throughout the book. Cool. 

Kim  14:09  
Okay, so part three then, Developing Multiplicative Reasoning.

Pam  14:13  
Yeah, huge domain that starts in grades three, four, and five. That is the land of, often, you know, three, four, five teachers are thinking about is that their job is really to develop multiplication and division. So, we define multiplicative reasoning and give lots of examples. But then also, what are the major relationships that lead to the major strategies both for multiplication and for division with whole numbers, with decimals, with fractions? Really helping teachers. Let me just stay on fractions for a second. Both in additive reasoning and multiplicative reasoning, we dive into the five meanings of rational numbers that are really important. So, shout out to Susan Lehman and her book Teaching Fractions and Ratios for Understanding. We dive into those five meanings to help teachers really build the sense and meaning of fractions before they dive into operations. But also then diving into operations and how the major strategies are showing up in whole numbers, and decimals, and fractions, and really thinking about how multiplication is connected to division and the two meanings of division. So, lots in those sections. 

Kim  15:17  
Heavy chapters. 

Pam  15:18  
Yeah, heavy chapters. You know, that three, four, five teachers are going to really want to focus on, and dive into, and get a lot of help. I think one of the questions that we get asked really often, maybe especially from fourth and fifth grade teachers, would be, "Okay, but yeah, I can see how these strategies work for some numbers. You know, for those nice problems that sure, sure I can use these strategies. But, you know, for most problems, we're going to need the algorithm to be efficient, you know, to be able to solve those problems." Well, I invite you to dive in and check out to see. Actually, give me any problem that's reasonable to solve without a calculator and with those major strategies that we develop in that part three... Well, and in part two, addition and subtraction. Addition, subtraction, multiplication, and division, give me any problem that's reasonable to solve without a calculator, and we've got a strategy that can get a solution as efficient as the traditional algorithms, but most of the time more efficient. And the best news is, while you're using that more efficient strategy, you're also developing either additive reasoning or multiplicative reasoning to be able to then continue to grow your math brain. So, it's brilliant. Yeah. I'm really excited about that. 

Kim  16:24  
Yeah. And I just want to chime in here that, you know, while these are you can go do in your classroom, it's also perfectly reasonable for teachers who want to grow their own numeracy to dive into these books, and tinker, and read, and play with the relationships and the problem strings that you have. You know, we always encourage people to just dive in and do what you feel comfortable with. But even if you don't feel fully ready, grab this book and start thinking about what you can use.

Pam  16:53  
Well, and you're reminding me, one of our features in the book is a Try This section. 

Kim  16:57  
Yeah.

Pam  16:57  
And so, we'll talk about a bit, and then we'll say, "Try this," and we'll actually give you problems to solve. And if you're doing it with a book study group, all the better that you can talk about your strategy and other people did it. Hey, one of the other things that are in both part two for additive reasoning and part three for multiplicative reasoning is also all the major models.

Kim  17:13  
Yes.

Pam  17:14  
For this strategy, what model do you want to use and why? And so, that's also in there. Yeah.

Kim  17:18  
Yeah. Okay, earlier, when you asked what parts of the book do I read. As I looked, as I remembered the part four title, I was like, "Oh, yeah. It is called "Putting it All Together," and maybe I never even read that. You would think I know every word. Part four, Putting it All Together. It's a great title. 

Pam  17:38  
Well, I'm glad you like it. That's excellent. You just skipped right over that when you're reading. That's funny. 

Kim  17:42  
Probably.

Pam  17:43  
So, yeah. How do we put it all together? We've given you this introduction. Now, we've walked through what are the major models and strategies for additive reasoning and for multiplicative reasoning for grades three, four, and five. Let's put it all together with a chapter on tasks. What are the kinds of lessons that you can use in your classroom? And this is where we parse out... Oh, and there goes my alarm to remind me.

Kim  18:04  
Love it. Love it.

Pam  18:05  
Can you hear that? So.

Kim  18:06  
I can. 

Pam  18:07  
I'm going to take a med while we're on this podcast.

Kim  18:08  
Your phone likes the title too. 

Pam  18:10  
There you go. Alright, so. 

Kim  18:11  
That's hilarious.

Pam  18:12  
Should we pause? 

Kim  18:13  
This is real life. This is real life. 

Pam  18:15  
You know, don't get old, everybody, when you have to meds. It has to be so long after you've eaten.

Kim  18:20  
I mean, wait, don't you want to get older? Like, that's a good thing. Like.

Pam  18:25  
Older and better. Okay, yes. More mature.

Kim  18:28  
Beats the alternative.

Pam  18:30  
Getting like Benjamin Button? Are we getting younger? What is the thing here? No?

Kim  18:34  
I mean, if you're not getting older, you stop aging. That's not ideal.

Pam  18:41  
Gotcha. That took way too long, didn't it? Sorry. Get old, whatever you do.  Too busy taking meds. Alright. We just get up close and personal in the podcast today. Whoo! So, in the Putting it Together, part four, one of the chapters is all about helping teachers understand the different kinds of tasks that really work well. So, we talk about Rich Tasks. We also do what I'll call a walk-through with those Rich Tasks. So, you get a chance to kind of feel what we mean. Sometimes teachers will say, "Oh, you're doing that discovery thing where you're going to send them off to, you know like, figure it out on their own. And once they've kind of messed around with some manipulatives, then you're going to tell them how to do it." So, that's not what we mean. And I can understand why that might be your only kind of guess about what we might mean by a Rich Task. But actually, we're going to walk through what it looks and feels like in two different classrooms with two different teachers. And a shout out to Kim because she's one of the teachers. So, we have some video of her doing a division task with kids. Holly Gabrielson Thiede, wonderful person who we worked with in the past, wonderful teacher has a Problem String for multiplication with a fourth grade group. And you really get a chance to feel what the kids are doing when we when you guys launched these Rich Tasks, and then had a Congress with them. Really nicely done. And so, again, get a feel for those Rich Tasks. Also, we define Problem Strings and get into more detail about what they are and aren't. We also talk about routines, and our favorite routines in grades three, four, and five. Shout out to I Have, You Need, which was Kim's invention. We have a whole section in there about how to use it, and extensions, and iterations of using that particular routine and some other routines. Besides the chapter on lessons that I just talked about, there's also a whole chapter on models and modeling, which I think might be one of the most confused bits of math education these days. It's not just about manipulatives, but it is, and the best use of manipulatives. But also using models that are really good to understand the situation and what's happening in the problem, aren't necessarily the same models as tools for building relationships, aren't necessarily the same models that are really good for computing. And so, we dive into what the differences of those are. We have some Venn diagrams about helping sort of separate and get more clear on how to use models to help students understand the math, but also to develop mental models in their heads, mental maps about the relationships. So, really excited about that. And then, lastly, in the part four, we have a chapter called Moving Forward. And I'm kind of really happy about this chapter. If you push me on one more thing that I was really happy about, I might have said that we kind of split it into three descriptions of readers. And so, you might, when you read the book, you might be ready to shift some of your teaching and your thinking, and so we give you some ideas of things to do if you're ready to shift. But you might be ready to turn. So, shift is like this little shift. And turn is like you're turning. And we give you some maybe more adventurous things to try. And, you know like, "No. Like, I'm ready to turn. Like, give me some things to do. You might be ready to dive in. You might have read that, and you might be like some of the folks we work with. Like, Susan comes to mind, who, once she sort of heard what we were doing, she just dove in, hook, line, and sinker and was Math is Figure-Out-Able, you know, 180 days out of the school year. And so, in each of those sections, we give you some bite-off-able, manageable things to go do, find successes, and then next places to go. So, I'm kind of excited about that idea of moving forward with next steps that are actually doable for you in your classroom with your kids.

Kim  22:33  
Yeah, yeah. And whatever amount you're ready to do. Because it doesn't have to be all or nothing.

Pam  22:38  
Right, right.

Kim  22:39  
Yeah. Yeah, yeah.

Pam  22:39  
Yeah. 

Kim  22:40  
Well, I mean, if you've ever listened to the podcast, you know that we have all the words about everything. But this is the book that puts it all together. This is the book that I wish I had twenty years ago when we first met. I'm so happy that, you know, on this journey, you've kind of like compiled it, and summarized it, and kind of thought through all the things. I think this really unpacks the most important things that you need to know for your classroom as a 3-5 teacher. And I'm thrilled that it's out in the world, and I am patiently waiting for my copy.

Pam  23:15  
Well, so am I. Mine hasn't shipped to my house yet. As soon as I do, I'll sign it and bring you one. 

Kim  23:23  
Okay, that sounds good. 

Pam  23:24  
Y'all, if you're listening and you're a K02 teacher, the K-2 book that has exactly everything that we just talked about but for kindergarten, first, and second grade teachers, make sure that you grab that one that's already out. If you're a 6-8 teacher, know it is coming. That book, we've already turned in the second draft. It is ready to come out September of... September? I'm pretty sure it's September of 2026. And the 9-12 book, we are currently writing, and so that will be coming out in about a year. So, just know that we have you in mind. We can only write so fast and get everything down. But yeah. Just, y'all, keep making math figure-out-able. We are so grateful for you and the efforts that you put into making your classrooms places where students are building and becoming mathematicians. And we thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, dive into the book that we talked about today, but also visit mathisfigureoutable.com. Thanks for spreading the word that Math is Figure-Out-Able.