Room to Grow Podcast

Season 6 Episode 9: Decluttering Mathematics five Fundamental Understandings That Unleash Meaningful Student Thinking

00;00;02;02 - 00;00;41;02

Joanie intro

In today's episode of Room to Grow, Curtis and I enjoy an exciting conversation with the author team of the book Decluttering Mathematics five Fundamental Understandings That Unleash Meaningful Student Thinking. The book will be available in late September 2026, but in this sneak peek conversation, you'll learn about what contributes to clutter in middle and high school mathematics and how to use big mathematical ideas such as measurement and equivalence, to get students thinking and making sense of what they are learning. We know you'll be inspired by these ideas from this amazing group of math educators, so let's get growing.

00;00;44;03 - 00;01;15;03

Curtis

Well, Joni, I am very excited for our conversation today. I'm not sure I've ever started our conversations without being excited, but I'm particularly excited today for our podcast because we just really have a great lineup of guests here today to talk about their new book that's coming out reasonably soon.

I think we'll be able to talk about that maybe a little bit during the podcast. But yeah, the conversation today is going to be just fantastic. So why don't you give us a heads up about who's on the podcast today? 

00;01;15;17 - 00;02;17;19

Joanie

Absolutely, Curtis, I'm really excited to and I just have to say, I knew Curtis was going to be really excited for this line up. When I saw the author lineup of this book, I came across this book and knew I needed to order it and read it, and thought this author lineup would be great for a podcast conversation. And also because Curtis and I are huge fans of all of these people's work. So without further ado, today on Room to Grow, we are welcoming the author team.

While most of the author team, 75% of the author team of a new book called Decluttering Mathematics five Fundamental Understandings That Unleash Meaningful Student Thinking. How's that for an incredible title? I'm sure y'all wordsmith. That too is very, very good. So let's start with some introductions. And like I said, we have three of the four authors I would love for you each to share with us your name, just kind of where you live. And then what are you most passionate about when it comes to math, teaching and learning? So let's start with Ted.

00;02;17;07 - 00;02;38;05

Ted

My name is Ted Coe. I'm coming to you from Scottsdale, Arizona. And I think you know what I'm most passionate about when it comes to teaching and learning. Mathematics is focusing on meanings and shared meanings and understandings. So I want to make sure that when when we're talking with each other about the mathematics, that we're meaning the same thing, that we're thinking the same thing

and what does it take to get there?

00;02;38;05 - 00;02;41;25

Joanie / Curtis

Awesome, great. I love that, Scott. You're next. 

00;02;42;05 - 00;03;22;25

Scott

I'm Scott Adamson from Mesa, Arizona, and I teach at Chandler Gilbert Community College, and I'm most passionate about being in the classroom with students, having just launched a new academic year this week, it's just been so fantastically fun to be in a classroom, getting students actively engaged in doing what Ted said, making meaningful connections and understanding and as our book title says, unleashing student understanding and student thinking. And so that's what I love to do as they productively struggle to do that together in a community at the whiteboards. 

00;03;23;05 - 00;03;29;25

Joanie / Curtis

I love it. Oh, so many good phrases, starting with loving the start of the school year. Very excited that you're starting the new school year. And April.

00;03;29;29 - 00;03;59;02

April

Yeah, I made April Strong. I also teach at Chandler Gilbert Community College, and I'm coming to you from Phoenix, Arizona. And I just have always loved mathematics and always wanted to be a teacher, actually. So I've spent my entire career teaching mathematics at the community college, and I just get so excited, like Scott, working with other students and just getting them energized and interested in learning math and wanting to know a little bit more about mathematics.

00;03;59;04 - 00;04;42;23

Joanie

Wonderful. Well, thank you so much for joining us from Arizona. I'm in Colorado, so y'all are my neighbors and we're really excited to have you on. I just want to call out to the fourth author of your team, James Tanton, who wasn't able to be with us today, but we definitely want to acknowledge his contributions to this book and to this important conversation.

So let's go ahead and just kind of jump in. I want to ask about the title itself, that decluttering mathematics, that's what kind of jumps off that cover image. And I would love to have one or more of you reflect on why you think middle and high school math need decluttering, like, how are you making sense of that and how did that drive the work of what you put into the book?

00;04;43;10 - 00;06;07;17

Ted

Hey, I'll jump in on that. First, the idea of saying, looking at the mathematics that's cluttered, what's what's going on here? Sort of. If I can point back to a time when one of my daughters was in middle school and she brought home some homework and I was looking at it and it said this, this homework, like, why are you doing this homework? This is not in our standards is you're not supposed to be doing this. And and then to further see that this was this was linked to proportional relationships. And I'm like, wait a second. This has nothing to do with proportionally they should ships allow me to be critical. And then and then to flip open the book and go, oh wow. You know, it's like professional relationship. You go from section to section and it means this here. And it means something different here to mean something different here. And it means something completely different here. And it not actually connected here. And, and to just go, oh, this is this, this is an example of something this just there's this, there's this clutter and then and then go well okay.

So we know that the content gets cluttered, that it's like these. Instead of looking at a puzzle where all the pieces are put together, we just keep throwing pieces into a box and having the students collect one more piece. Right. And so and so from a cluttering perspective, we're like, oh, there's all this clutter in the math content. But more to where we focus the, the, our energies in the book is to say the result of that is cluttered thinking. And what can we do to try to find these fundamental understandings that help to declutter a thinking that is the result of all of this other mathematical clutter that's floating around? 

00;06;07;29 - 00;07;43;00

Scott

And just to give another example, I recently interviewed some elementary students.

Some fifth grade students. Asking them questions started with a question about multiplying two fractions. And here's a great example of clutter. I saw all kinds of craziness. Now, of all the all the procedures that one just multiplying straight across, if all they knew is the procedure, you might think, well, that that would be a natural procedure to do. But oh my goodness, they wanted to cross multiply. Because to Ted's analogy, that's one of the puzzle pieces they learned once completely disconnected. Just one of the pieces that was tossed into a thousand cases, a box that had no connection, no to anything meaningful. So they tried that some. Of course, you worked with elementary students. You might know this. They said, well, first you have to get a common denominator, because apparently, to the mind of a student, if there's fractions involved, you must always content on nominator things with common denominators. And they mix together procedures, you know, from this thing and that thing, it was an utter mess. And so that's a great example of students coming to the mathematics from a cluttered perspective of of grabbing puzzle pieces randomly and trying to throw them at the problem.

Right. Instead of thinking, we say in the book, one of my favorite things to say is we think students should think first. Compute only if necessary. 

00;07;43;20 - 00;09;04;21

April

That's one of my favorite sayings to actually Scott. But I want to jump in and say the three of us. And also James has been doing and working with professional or with teachers in professional development for many, many years and in working with teachers.

What we noticed is that teachers are just bombarded with a bajillion things that they have to teach. They've got these very large textbooks with all kinds of content. They have their standards, some of which don't necessarily jibe with the content that is listed in their textbooks. And so teachers often just teach all the things because that's what they think they are supposed to be doing.

And so what on the receiving end for students, what it comes out to be is a bunch of puzzle pieces like we were just talking about, and they're all disconnected and they don't even fit together in many cases. And as I kind of jokingly say, sometimes there's often missing pieces to the puzzle. So as they're trying to put some pieces together, they don't have all the pieces to actually connect, and maybe other extraneous pieces are dumped into this box too.

So it's like a literal jumbled mess in student's minds. And the teachers need some rethinking, actually, of the content that they should be teaching in our world. 

00;09;04;21 - 00;09;59;00

Curtis

I love that you guys chose. So while we're on the topic of the title, so the decluttering of mathematics, but the term unleashed that Scott stopped on it earlier. I love this term unleashed and unleashing meaningful student thinking that concept.

So I definitely hear the the statement and we've we've talked about it maybe before on the podcast, kind of this idea of so many things and so many choices and all of these things kind of. And I love the analogy of the puzzle box. That's, that's so that's so good, so good. So many little ties there. But this idea of unleashing, how does that happen?

So as we do this decluttering, as we talk about what that means and how people are going to navigate it, and we'll get to some of those questions in a minute. But what does that mean for student thinking? How come you chose unleashing meaningful student thinking?

00;10;00;17 - 00;10;57;20

Ted / Joanie / Ted

I think it starts with my little phrase before is we want our students to think first. I love that unleash the thinking. And it's tied to these fundamental understandings. So what are they thinking about? Yeah. So we we argue in the book that what they're thinking about are five big puzzle pieces. If you can think, make connections, make meaning of these five big ideas, all the other things will flow out of it. So instead of 1000 or 2000 individual tiny, tiny, tiny little procedural puzzle pieces, we argue that there's five. Not that there's only five. We're you know, we only wrote the first. Maybe there's going to be more

 Once we got these five down. Here's what these five down. Here's what's next. 

Yeah. So the we unleash their thinking in related to those 5 fundamental understandings, which then lead to how I'm going to approach this, this problem situation, which then may lead to some procedural doing as it makes sense to do.

00;10;57;28 - 00;11;14;23

Scott

So I appreciate yeah, yeah. Drew out of these out of what we've seen working with students and with teachers over the course of our career where it's like, oh, wait a second. There's some ways of thinking that if you just if you just played these out first, the approach to how you're going about this problem is going to be completely different than just randomly grabbing a piece.

00;11;14;28 - 00;12;09;00

April

Yeah. And I was going to jump in and just add to this. I love that you picked up on that word unleash. Truth be told, we actually paused quite a bit on on that word to make sure it was doing what we wanted it to do as a part of the title of the book. And when you think about students experiences in class, thinking often doesn't happen.

But it could, and I I'm a firm believer, even after almost three decades of working with students, that students will come to class genuinely wanting to think and wanting to engage in meaningful discussion and different things. But we don't often have those opportunities in class for learning mathematics in ways that get the thinking out there. And so we really wanted our our book to communicate to the world that student thinking should be unleashed and on the teacher. And what can we do to help that happen in a class setting.

 I love that.

End of Segment 1

00;12;09;15 - 00;12;19;00

Music break

Start of Segment 2

00;12;19;01 - 00;13;14;01

Scott

If I could take it, take us back to my example earlier. I think it just connects so well. The problem that I gave these fifth graders to multiply two fractions was something like one half times 6/5. So imagine instead of just, oh, what do I do? Which puzzle piece do I grab when I multiply fractions? Which procedure. Yeah, if they cross multiply whatever.

But what if they thought first? What does multiplication mean? What does it mean to say I have 6/5. So what's the meaning of 6/5 six copies of one fifth? And then what if you just thought about what half of 6/5 would be? Well, if you have 6/5 and cut in half, you have 3/5. So there really is no said my my phrase calculate only if necessary. It wasn't even necessary to do a calculation. You just had to think about what half a six would be, right? I don't see that. We just don't see that.

00;13;14;24 - 000;16;23;15

Joanie

And I just think about the excitement that would generate in students. Just your phrase, Scott, like, I can imagine a student in your classroom who hears you say compute only if necessary.

Like, man, I can think of so many kids, middle and high school age that would just light up at that. Like, wait a minute, I don't have to compute all the time. And that to them it would feel like, yeah, oh, I found a shortcut or I found a trick to, you know, not having to go through the steps of math, which, you know, I'm circling back to, to April's comment, like, I totally agree that kids come in wanting to think they want to understand. And when we just go to like, here's how you do this kind of problem. Step one, step two, step three. And I'm saying that critically, fully acknowledging that at the start of my teaching career, that's exactly what I did. That's what I thought it meant to teach math. So having an alternative to that for teachers to latch on to. And I just want to also throw out this idea of reducing it down to five fundamental understandings. And I appreciate you saying that's not all that's important. Certainly. And, you know, six years of seven years of of math learning for students, but five is tangible, five is doable. Five is like, okay, I'm no longer overwhelmed by the laundry list of standards or the 17 chapters in my textbook that I that I feel like I'll never get through and help students make meaning of. And then just one last thing, and I would love you guys to respond to this. And if there's any connection to this in your book, one of the things that I find myself saying over and over again to people who maybe don't have the background of studying mathematics deeply, or teaching mathematics, which I think when you teach mathematics, you learn it in a whole, at a whole different level than just, you know, knowing it. Right. But there's this misconception that math is a stair step, right? Like we hear all the time, oh, these kids that are starting algebra one, they don't even know their multiplication facts. Like, how can they be expected to learn algebra one content when they don't have the basic skills? So there's this belief that you have to master this step before you can move up to this step, and you have to master that step. Whatever, quote unquote mastery even means to go up. And, you know, as a 20 years teaching high school math, it wasn't very far into my career that I went, wait, it's not a stair step, it's a spider web. It's like there's all these different ideas and there's connections between them. And even now, you know, I started teaching in 1989. We're talking several decades later. I'm still making connections between mathematical ideas that are new and exciting and interesting to me. So I'm curious if any of those concepts came come out in the book of thinking about connections between ideas, or debunking this myth that students have to master something before they can proceed to deeper thinking.

00;16;23;15 - 00;17;08;29

Ted

I think yes, maybe not exactly that question, but the idea of, for instance, one of the fundamental understandings that we write about is measurement. And in our book we take measurement from even though we targeted 6 to 12, we probably even take it back before that. But then you see the idea of measurement and how it it goes through all the way into trigonometry. So we try to make those, those connections and that maybe spiderweb of how does measurement play out. And if you have the fundamental understanding of measurement from an early on learning experience, then that idea of measurement then persists through so much of the of the mathematics that students learn in grades six through 12.

00;17;09;02 - 00;18;49;20

April

Yeah. And Joanie, I love your spiderweb analogy. I think that works so nicely. But students also feel very much like they are required, if you will, to do the stair step idea. Yeah, just yesterday. Just yesterday, I was teaching a class in calculus two at our college, and it came up about the natural logarithm. And so, so a student was like raised his hand and says, I'm embarrassed to ask this, but does a natural log always have a base E?

And so we took a moment. Now, this is, of course, content that in theory, one should have mastered before they got to calculus two. But now here's a moment in class where I think to myself, we're going to talk about this. Let's just it's not really content in that class per se, but let's just take two minutes and let's talk about natural logarithms and logs in general, what it is, what it isn't.

And I invited the students to kind of just, you know, with these sort of topics come up that you, in theory, have experienced before, but maybe totally no. Maybe not. No. It's okay to bring it up and let's talk about it, that it's that spider web that connections are made. Perhaps later in your math journey, not when it's actually just taught for the first time.

And I think that's important for all of us to recognize that it is a journey. Math is a story that unfolds over a number of years as we engage in more and more mathematics, and students down the line will make connections that they had learned of mathematics years prior. And I think that's an important thing to embrace. 

00;18;49;20 - 00;19;13;20

Joanie

Yeah, I love that as a frame for debunking that they have to have some sort of mastery of something before they can do deep thinking, because you just exemplified April, that being able to engage in the deep thinking is actually what's contributing to the mastery, the really deep and robust understanding.

Like you can't, it's not one before the other. They have to be working in tandem. 

00;19;13;20 - 00;19;57;20

Scott

And I appreciate, Johnny, your, your, your web analogy. It really is the schema of ideas, right, that we have coming together. And that schema is different than like you might see how standards are connected to each other in a grid. So, so the idea that maybe when you when you my concern is when people talk about like standards being relentlessly hierarchical or something that that that implies that, well, if they're not getting this standard then I have to go to the next nearest standard and then I have to go to the next. But that's not that's not the case. The case is there are these fundamental understandings that are lurking behind the scenes in this cloud that unlock all kinds of connections. If we if we just if we just allow ourselves to break out of that step by step mindset that you were talking about

00;19;58;15 - 00;21;34;11

Joanie / Curtis

 All right. I'm tempted to ask. Sorry. Go ahead Curtis.

It's okay. I just I love the terminology that is being used around this, this idea unleashing unblocking un un unbreak breaking free from sort of this, this restrictions. Right. That we feel the students feel the teachers feel pressured by all of that content and even that interlocked web that you mentioned, Ted, of, you know, we've seen the connections between all the standards.

We've seen that picture right before that there's some sort of a piece here. But the fact that you bring up that there are ideas, big ideas lurking behind that, that we can tap into to give students freedom. And in my head, I'm thinking of my ten year old, now 11 year old son. Just the freedom. If he is in a space to say, here's how I think about this.

Like, let's just let's just think first and calculate if necessary, I love them. I'm so good. I mean, my wife's going to make us t shirts with that on it. It's going to be great. Yes, we're going to be using that. But I just love this concept of freedom. And what's happening in my head is thinking about the freedom that students who are in classrooms where this becomes the focus or becomes the culture, the freedom to think and the freedom to ask the question, I'm sorry, but log natural log. Is it really always basi? Is that a thing like the freedom to ask that oh so awesome. Yeah, fantastic.

00;21;35;00 - 00;22;46;23

Joanie

 Okay, so I, I'm so anxious to read this book and I'm finding myself sitting here going, okay, I love that there's five. We're talking about these, you know, big nodes of understanding hiding behind the clutter of mathematics. I need to know what are the five?

What are they? I'm really curious because and let me frame this for our listeners. Like it is just my own curiosity. But also when I think about grades six through 12, right, like you guys are suggesting, like here are five ideas that are big enough to encompass, you know, a whole like half of a student's schooling in mathematics. And, you know, I want to be able to think from I'm a sixth grade teacher all the way up through, I'm an AP calculus teacher. And how these these five big ideas will resonate to help me kind of manage, like all the extra stuff that's in my textbooks or in my state standards. Like how do I think about these five within those contexts?

So I would love to hear I would love to have you guys expound. And we probably don't have time to go deeply into all five, but I would love to hear what they are. And then maybe we can expound on a couple of them. 

00;22;47;03 - 00;23;53;09

Ted

Well, Scott, you already mentioned one of them. Measurement. Yep. There's number one. Oh, I don't know. It's number one, but it's the first song we talked about. It's one of them. It's the one that we talked about first is measurement. I'll, I'll offer a couple and I'll pitch it to the team comparison in the sense of there's additive comparisons. When I say I have some money, you have some money. How much more money do I have than you?We're just going to compare additively by subtracting. But then you might say you might compare multiplicative or April shall I say multiply. Yeah.

We might compare by saying I have twice as much money or half as much money. And so getting into those comparisons and then as I mentioned with measurement before, if you have a strong fundamental understanding of comparisons that plays out throughout linear relationships, exponential relationships, trigonometric, it just plays out and connects to so many things. But it all goes back to this foundational understanding of what does it mean to compare?

00;23;53;12 - 00;23;59;17

Scott

Yeah, percentages become trivial when you have a good, powerful way of thinking about multiplicative. You don't need any percentage formulas, as it turns out. 

00;23;59;17 - 00;24;20;17

April 

And yet, what I often see when I go into classes is the proportional relationship developed by formula of the is over of equals, percent, over 100 to just solve things like what is 15% of 24 rather than just thinking about it multiplicative even.

00;24;20;19 - 00;25;22;18

Ted

Which by the way, is a third fundamental understanding proportional relationships. You know, I was expecting that one. Maybe they're not me from the cross multiply. In fact, that would. That is the only place that cross multiply comes up in our book is we have a feature in every chapter where we suggest, or do we even do it more strongly that do we demand that certain things in mathematics be placed in the mathematics attic? Oh, nice. And the old butterfly method cross multiply. We say put that in the attic as far back behind identical junk. Now there's some things that we say put in the attic because it's historical. Maybe it's interesting, but it's just not necessary in this world. And so sometimes it's fun to go up in the attic, grab all things, dust them off and look at them and remember the old times.

But there's still things that should be put in the attic forever. So proportional reasoning, scaling in tandem. 

00;25;22;25 - 00;25;46;15

Scott

Ted. That's right, that's right. So the idea of what is a proportional relationship, it's it's simply in this case we're going to say we can start off with you've got two quantities that scale in tandem. If you double on the other doubles, if you 111 you triple other. And when you start to come at things with that perspective, you find it everywhere, right? And from there you can do you can do the thing first, compute when necessary. And it turns out that all kinds of things just fall together. Yep.

00;25;46;15 - 00;27;49;13

April

 Yeah. Another one of our fundamental understandings is a rethinking about graphing. And so traditionally graphing is a let me just plot points and I'm going to connect all the dots and boom, there's my shape. There's a thing in mathematics education research that distinguishes between developing students thinking about graphs from a shape perspective versus a more dynamic perspective. So we don't want students just memorizing shapes of graphs. We want them to actually be able to covariate these two quantities that a graph is illustrating and walk away. Thinking about the graph is this image that's produced from this covariate relationship of two quantities. That is such an important image to develop with students. This more dynamic rather than static image of a graphing is. And that just takes students all the way through all of mathematics. We were just developing this idea and calculus this week in kelp one specifically, and I mentioned to the class, if only your experience when you first started with graphing was like this, we wouldn't have much to talk about right now.

But what you do because you've just memorized shapes, there was a moment where there was a graph up on the board that was concave up, and so it didn't want to talk about concavity in that moment. But a student says, oh, this is accelerating. The runner was accelerating. Well, it turns out the axes were switched and time was not in the horizontal axis.

It was on the vertical axis distance. And so a concave up in that moment when you are graphing distance and time where time is on the vertical concave up graph doesn't show that the runner is accelerating, but that student had just memorized this means this thing, and they attach labels to certain shapes. And that doesn't contribute to real thinking and understanding.

00;27;49;13 - 00;29;16;16

April / Ted

So grasping in general is one of the chapters and one of our fundamental understandings that we talk about and how to really pull that out of students from a sense of meaning and thinking, rather than just memorize a shape and put a label and just tease a little bit on this to when you read this chapter, one way that we help students to reason in this way is to think about what we call calculus triangles.

And by the way, we call them calculus triangles, even if we're in an algebra class. That's right. We don't care where we are. They're still calculus triangles. But the idea is changing the whatever quantity we're keeping track of on the horizontal axis, changing it in uniform amounts, and thinking about what would the chorus finding change in the vertical quantity B is related to that.

So you can see sometimes students will say because they have this prior memory. Oh, you mean like rise over run. And I said, well, what do you mean by over rise? So April, this week in my class, one student, as we were doing the different activities and creating these graphs based on some context, the student says it's really helpful to think about the calculus triangles first and then put the graph in.

And I was like, yay! Yes, thank first. Thank you only when necessary. Yes. So it's become a in this class. Now it's become kind of a taking a share. This is a big thing. The calculus triangles first and then the graph. So, there's four, four out of five.

00;29;20;22 - 00;30;37;16

Scott

I got I got it I got it. The fifth one equivalence, equivalence. We lead off the chapter I love talking about this. We lead off the chapter with this. In elementary school students might see three plus two equals. What does equals mean in that case to the student equals means in that student compute three plus two equals. And the only thing that they could write is a five. So we perturb that.

We put three plus two equals one plus blank. And they're like whoa whoa whoa three plus two equals one. We they missed the whole point three plus two equals one plus four. And so that leads into algebraic representations two x plus six equals five. What does that mean. That leads to the so-called simplifying of expressions which we're not really a fan. We're putting that in the attic by the way simplifying simplifying. Yeah. What is what is simplify mean. We use the example of a of a radical expression some crazy, which is kind of ridiculous in the first place. You have some x to the y to the some things to the something that square root of all that. And why in the world are really doing that?

And what does it mean to simplify. And so we, we tackle that issue in that in, in that chapter, the fifth fundamental understanding. 

00;30;38;00 - 00;30;38;28

Joanie

Amazing.

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Music break

Start of Segment 3

00;30;48;20 - 00;31;40;01

Joanie

Okay. I'm loving I'm loving these five. And my head is running around a little bit in terms of what these mean I would love I'm so I want to I want to maybe just pause a little bit here and ask you to elaborate a little bit more. So I'm, I'm hearing some applications that I can connect to high school content.

That's where my teaching experience is. I taught for 20 years in high school math. Can you help us, like pull out some specifics? And I think you did it a little bit with the proportional reasoning, but also with some with the other topics specific to some middle school content. Give us that. Like how does focusing on this fundamental understanding in middle school replace the clutter of what's what's typically in the middle school curriculum?

00;31;40;03 - 00;31;54;02

Ted / Scott - then everyone briefly

Scott I'm thinking of the story of X, so I was just thinking that's what I'm thinking that. So yes. Tell the story. Yeah. Tell the story of X. We have time for plenty of. Yeah, we have a little time. The story is what it's all about. 

00;31;54;02 - 00;34;10;06

Scott

So back to that chapter. We're talking about what is equal mean. And that transcends probably to a middle school topic of we say solving equations. So we do it something like this. Imagine two x plus six equals nine. What students often do at we're trying to fix it is they approach that from a very procedural perspective. I have to do things to both sides of the equation and things like this. We approach it from the story of x means this. When you say, I forgot what I said, exactly what two x plus six equals nine.

What's the story of x? Oh, there's an unknown quantity x that's being multiplied by two and then increased by six to produce a result of nine. If you can articulate the story of X you can undo. We call it the the deconstruction. The construction is the story of X, that deconstruction is well, if you multiply this by two and increased it by six to get nine, well then you can discern then that two x has to be three.

Because if you have a quantity that you increase by six to get nine that quantity has to be three. So we actually ask students we call it preserving troops. Oh nice. It's true that two x plus six equals nine. Then it just follows that that quantity two x has to be equal to three. So we just we think and we write true things. Yeah. Preserving truth. And you can just keep writing true things. And till the result that we all want is what is that value of x becomes just obviously true. It might even be true. Let me let me fix it just a little bit to, to make my point, if we said two x is equal to four, you could maybe stop there because most people are going to look at two x equals four and go, well I must be two.

Like is that enough truth in that moment to illuminate what the unknown value is? Or if it's a little messier, do you need to go one more step and just be so clear that x has the equal to right. But we're just laying down using the story of x. We're laying down through statements preserving truth. That's what algebra is all about learning truth.

00;34;10;07 - 00;35;02;00

April

Oh yeah. And that really contributes to this notion of what the equals means that you are literally writing and rewriting truth statements. That makes sense. And students then in your story of X Scott students then when they solve your equation two x plus six equals nine and they get an answer, they can produce a reason for their thinking about how they rode through this, not just a okay, my teacher told me subtract six here, subtract six here. The next statement. Then divide by two divided by two. Ultimately, they will do some of those things once they've established a way of thinking through this equation. Solving using equivalence ideas and using this rewriting of truth statements. And I think that's a really productive way of thinking about solving equations. 

00;35;02;07 - 00;35;49;10

Scott

And that and that moves along kind of back to our story about the spider web and how these fundamental understandings of the leash thinking when students in our classes have had this kind of instruction and our thinking about the story of X, when they get to more complex equations like, let's just say a radical something with a radical in it, square root of blah blah blah. Plus who knows what equals whatever. We don't even have to show them, teach them, tell them whatever how to solve it. We just say, what's the story of X? Okay, then make some more truth statements. Undo. Deconstruct the story of X. We just get out of the way and students salt radical equations in moments because they're thinking they're not just doing the grabbing puzzle pieces.

I don't remember how to do radical equations. Well, it just just use your brain and think you'll figure it out. And so we in terms of one of the one of the questions we often get in this is how do you have time to cover all the content when you're doing all this deep thinking and active learning and so on for our little phrases, go slow to go fast, right?

You go slow and teach students to think. And the story of X is a fundamental understanding when you get to other things. And we could spend ten more podcasts on all the different quadratic equations. What about this? What about that exponential whatever. But when they have that fundamental understanding, the learning goes so much faster

00;35;50;29 - 00;36;56;23

Joanie / Curtis

Oh yeah. Oh yeah. Yeah, for sure. I can imagine even just this, this fundamental understanding driving some another one of the topics you brought up of graphing just this, I mean, just the connections you're even making here with the story of X and, and immediately when you were talking earlier about not just thinking about the shape, but actually the story of that function or the functional thinking that contributes how these two things cauvery. Wow, what a fundamental thing that maybe we as math teachers, you know, after we've taken so many courses and we've done and like Joni, I taught high school mathematics as well. But that that idea kind of maybe is something we don't think about that we do. Right. Like it comes sort of like that's something we did. We thought that way, even though maybe procedurally our teachers taught us to do things very in a box.

00;37;25;17 - 00;37;45;12

Curtis / Joanie

But our brains were kind of thinking in the ways that you guys are describing, like, I wonder how many folks, I'm so excited to read this book. I'm so excited to go and get this. This is this is one of those podcast conversations. And for sure, the book will have this effect where I'm like, I want to go back to the classroom.

00;37;45;12 - 00;39;44;16

Joanie

I want to teach this way. I want to go give me a do over, please. From all the things I didn't know how to do the right way when I was in the classroom, I want to just acknowledge that we're going to have I. I'm confident, 100% confident that many of our listeners are going to be like, yes, like this is about sense making, and this is going to contribute to my students own identity and agency.

And yes, I've always said we have to make sense of it before we put the formal language in processes around the mathematics. And this is just so perfectly in line with all of that. And at the same time there's this. Yeah, but where teachers are feeling the pressure to, you know, I have to put students will be able to on my board because my administrators can expect to see that when they walk in or I'm teaching, you know, my eighth grade math class with five other teachers in my department, and we all have to be ready for test day on the same day and follow the same, you know, scope and sequence in our courses.

So what what thoughts do you, the three of you have about how teachers can navigate like there is this wonderful sort of ownership and autonomy about being a classroom teacher, that you do have control over what you do and say and how your students engage. But we also are operating within a larger system that has some expectations and in some cases, some constraints.

So as I think about a classroom teacher who may be really excited about this, and this is kind of a breakthrough way of thinking of how they can shift their practice, how do you recommend that they manage some of those external pressures, maybe around parent expectation or administer administrator expectation or, you know, the the drive to standardize assessments or managing their curricular resources, like pick any of those. They're all challenges, you know, how would you recommend teachers navigate those to shift into this kind of way of focusing their teaching?

00;39;44;20 - 00;39;55;25

Ted then they talk over each other

Who's going first on that one? Yeah, it's a simple question. I'm sure you have a three word answer, but we do, we do, we do, we do, we do take this on towards the towards the end of the book.

00;39;55;26 - 00;41;54;22

Scott

Right. And but it's just it's this recognition that you're you as a teacher, you are being pulled in different directions, right? You have your idea of what it means to, to truly learn mathematics, but you also have all of these other pressures that are on there as well. And so it's not it's not necessarily an easy thing to navigate. You can't there's a lot of, you know, there's there's truth in what Scott saying about the go slow to go fast. If you really know this stuff, then they can fill in some of the puzzle pieces, right? They can think their way through it without. It's not just because you didn't teach them one particular procedure. That doesn't mean they're not going to be able to solve the problem, right?

You've got something bigger, but there's also some there's also some, some, some strength and not going alone. And maybe you turn maybe you turn this into a department level or a district level conversation. Right? You start to say, let's do what this and, and let's acknowledge some of these things, these other pressures that are out there. One of the, one of the one of the things that we tried to do when we when we started writing the book was we we held ourselves to the task of saying, what are we really about? What is it? What is it that we're really what does it really mean to teach and learn mathematics? And we and we landed on three, three things. And is it is it good if I share these now. Yeah. Yeah. So so so so the one is that we are about ways of thinking about mathematical understandings. This is this is a key thing that we need to do. And if that doesn't happen then we haven't succeeded in teaching ways of thinking paired with ways of approaching mathematical challenges. So the idea of how do students approach a problem? What am I helping them to think about? I'm not. The first question they ask shouldn't be how am I supposed to do this right? That the first question should be, what can I do here?

What? What are the possibilities? How can I think through this? Right. So ways of approaching these mathematical challenges, ways of thinking, those were our first two. And making sense and occasionally using some ways of doing right. Those things are still the procedures. These things are still in there. But all of these are in the service of doing bigger things.

00;41;54;23 - 00;42;18;15

Scott / April

And so if I'm not, if I'm not bringing together ways of thinking, ways of approaching ways of doing to do bigger things, then we're not winning what we're trying to do. That's so true. You know, I think it's also a rethinking of what we're trying to do with our students in a math class. Like we haven't mentioned this yet, so I'm going to drop this artificial intelligence in our world with AI.

00;42;18;15 - 00;43;36;24

April

At this moment, I think we are absolutely at a place where we need to rethink what we're doing in the class, regardless of these sort of external forces that might be pulling us in different directions as teachers. But if we can shape our thinking to be about teaching is really leveraging student thinking, then we're going to make some changes and we're going to be able to leverage what what Ted was talking about with ways of thinking to ways of approaching to ways of doing, to really pull out the mathematics with our with our kids. So, you know, I just Joanie, you had mentioned about putting up the objectives at the front of the room, you know, and and I know administrators come in with a checklist, check. Number one, are those objectives. If writing those objectives at the front of the board was going to solve math education challenges, we would have like been solved a long time ago.

We would have no problem to talk about now. There's crazy things that we tend to do. But regardless, I think, I think if we can just encourage our teachers to just focus on student thinking and have their instruction be a response to student thinking that's provided and giving students opportunities to to think deeply, I think will make some make some change here. Scott, what do you think?

00;43;36;25 - 00;44;39;22

Scott

I don't know how serious our teachers can take this, but when I have to write the objective on the board, what I, what I tell them is try this. The student will be able to think first and compute only if necessary. Yes. Put that up on the board every day. I love that, I love it.

I got the check mark for that. Yeah. And the other thing is kind of to tie together what we're saying here. I really have a strongly held belief that when we teach students to think and to reason and to have these bigger fundamental understandings, we don't have to teach all the thousand puzzle pieces that are going to show up on that assessment.

In fact, they don't remember the thing. I mean, my multiplication of fractions is a great example. They would have got that question wrong anyway. So if they're thinking they can see a problem on an assessment and think their way through it, I really believe that. And we have some evidence of our teachers we worked with that have higher standardized test scoring than or the district testing whatever than their than their peers.

00;44;39;22 - 00;45;01;02

Scott / Joanie / Curtis

And then there was a oh, there was a third thing. Oh, shoot, I lost it. That's all right. I'll think it is. Okay. Those were really good ones. You don't need a third thing. Those are those are really good ones to to hang our hat on. I know I'm, I'm sitting here thinking to myself, how in the world can I get my hands on this book?

00;45;01;02 - 00;45;40;25

Curtis

Because I'm, I have two. I get to more readers because I have my boys at home still. And so I get to to try this stuff out on, on on my two guinea pigs at home and high school students living in actual actual middle and high school students living in my home. So I get to try these things.

But I asked that question also for our for our listeners just thinking about, you know, I'm very excited about this book. I know others will be as well. So where are some of the places that we will be able to lay our hands on this once it's once it goes live? 

00;45;46;02 - 00;46;11;12

Scott / Curtis / April / Ted /Joanie

Well, Amazon already has it available for purchase even though it hasn't been released yet.

 So okay. Okay. Yeah. On Amazon, preorder it today on Amazon. Okay. And on website. We'll have it as well. Yeah. And it's by Teachers College Press. So on the Teachers College Press website it also is available for preorder. Awesome. We'll put all those links in the we'll put all those links in the show notes. And yeah, what the what makes sure we have all those links up there.

00;46;11;14 - 00;46;35;10

April / Joanie

Yeah. And if you're coming to NTM we actually have a workshop that we're running. So take a look at that and another talk as well. So hopefully those that are attending NTM can can join us. Awesome. Well thank you so much for the great ideas. I know you've generated a ton of excitement and Curtis and I for this book and hopefully a lot of others as well.

00;46;35;11 - 00;46;44;10

Joanie / ted / April

And we look forward to lots of opportunities to engage and learn from you. Thank you. Yeah. Thank you. This opportunity.

00;46;44;13 - 00;47;02;15

Joanie Outro

Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator.

See you next time!

End of Segment 2

00;21;46;15 - 00;36;56;00

Music break

Start of Segment 3

00;21;57;06 - 00;23;41;14

00;38;03;20 - 00;38;23;10

Joanie Outro

Well, that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us, consider leaving a review to help others find us and share the podcast with a fellow math educator.

See you next time!