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Joanie

The way this teacher framed it, it wasn't just factor these 20 and get the factor pairs. It was group them. Find a classification like as you factor these look for commonalities. And it became a sorting activity. Well that totally changes the kind of mathematics that students are thinking about when they engage in that task.

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Joanie

In today's episode, Curtis and I visit another of the mathematics teaching practices, from Principles to Actions, as part of our 2025 series honoring the 10th anniversary of this important and CTM publication. We discuss implementing tasks that promote reasoning and problem solving by carefully unpacking the meaning of implementing of rich tasks and of reasoning and thinking. And we end in a place we didn't expect when we started. So let's get growing.

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Curtis

Well, Joni, we are recording our what second podcast, I guess for 2025. Yeah, really excited to be doing this again with you recording the Room to Grow podcast today. And today we have a really exciting topic. I, I'm actually really intrigued by this. We've been talking a lot about just how this how this works. And for those of us, our listeners that are maybe new to the podcast or didn't hear last month, Joni and I are, working through Principles to Actions, the CTM publication this year, and as a way of sort of celebrating the fact that it's it it's in its 10th anniversary. Yeah. Which is fantastic, that this work has been out for those ten years. We're looking through and talking a little bit about the mathematics teaching practices, that sort of are the foundation of the Principles to Actions book.

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Joanie

Absolutely. Yeah. So we had a great conversation last month around, facilitating meaningful mathematical discourse. And I loved how we brought in, you know, ideas from the standards for mathematical practice and attending to precision and and really, that was a much deeper conversation about speaking mathematics, than just facilitating classroom discussions.

And I suspect that that's how today's conversation will roll out, too. We're going to be bringing a lot of additional thoughts and perspectives into this idea. But today we're going to be talking about implementing tasks that promote reasoning and problem solving. So how where does that make you want to start, Curt? 

00;02;37;15 - 00;04;08;15

Curtis

Well, the first thing it makes me want to start is by admitting the fact that when we were talking  about this in in preparation, much of my thoughts were around the selection of tasks. And I think actually, that probably is something that a lot of teachers think about when they think about this, this math practice. And this, this idea of tasks in the classroom. And, you know, I even I'll admit it, I thought, without revisiting the the book, I had been thinking about selecting mathematical tasks that promote reasoning and problem solving as sort of that math practice.

And in it, it goes far beyond the selection of. But, and I think I said this in our preparation conversations, earlier that, you know, the selection of a of a deep or a rich mathematical task is not an easy one. It's not easy to find and select those tasks. And it's an important process because in order to implement tasks well and in order to position them to promote reasoning and problem solving, it's easier to do that when you have a task that kind of sets you up for that. You can do it with a bad task. Well, that's a bad word. I shouldn't use that. You can do that with a task that maybe isn't set up for reasoning and problem solving. Certainly a teacher with a lot of skills can can take a task that's maybe not set up that way and do it in a, in a, and hold fidelity to this, teaching practice. But it's easier when you select a good one. 

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Joanie

Right. And I think, you know, coming back to the fact that Principles to Actions is ten years old now, I think about ten years ago, I think there was less understanding of what, or I mean, less than now. But even, you know, ten, 15, 20 years ago, I have I have a lot of years under my belt in this space.

And I think there had been some research that is a couple of decades old now that really talks about this idea of high level. I mean, there's a lot of catchphrases that we use, right, like high level tasks or rich mathematical tasks or, you know, there's a lot of things that we hear, but, and there's some, some great connection to the research from Smith and Stein that dates back to 1998, in principles to actions around, you know, classifying tasks, whether they are low level demand, high level demand, and kind of how to work your way up that hierarchy. But I think, you know, to your point, it's not just about task selection. So I'm thinking back to, you know, when I was in my district math coordinator role in working with educators, I saw a really brilliant educator take a worksheet that was basically, you know, 20 practice problems, a skill maybe it was, let's say, factoring trinomials. And it was literally here's 20 different trinomials factor them all that wouldn't fit Smith and Stein's qualifications of a higher order task. Right? There's not multiple entry points. And, you know, many, paths to solutions and requires use of multiple representations like that worksheet was designed to be what we fondly called kill n drill. It's just meant to be a practice worksheet of practicing a procedural skill. But the way this teacher framed it, it wasn't just factor these 20 and get the factor pairs it was group them. Find a classification like how do you factor these. Look for commonalities. And it became a sorting activity. Well that totally changes the kind of mathematics that students are thinking about when they engage in that task. So it's not always about the task.

Although I agree with you, when you have a task that allows for multiple entry points and at different ways of solving and you know, and engages students in multiple strategies, those certainly lend themselves to these higher order thinking, but it's not a requirement. I can take something as simple as a practice worksheet and implement it in a way that gets students to think about and engage in sense making and problem solving and reasoning. So I think it's important that we call that out. Task selection is important, but it's not the end all be all. 

00;07;04;00 - 00;08;38;07

Curtis

Yeah. And what I what was going through my head is you were talking about that and I'm imagining this, this worksheet of 20 Trinomials. And I need to factor them into their pairs. And then trying to think about what the grouping might look like. Because there's different ways that I might, I might choose to group them, and I might have reasons for why they're grouped certain ways. And that's a really that becomes a very broad task. But you know, what's, what's constant is that I'm still getting some practice with that procedure. Right. And one of the things that we hear, you know, when we talk about these rich mathematical tasks and we hear about all of the different kinds of, of teaching and trying to go for this conceptual understandings and trying to get at, at, mathematics and the connections that we're trying to draw and help students to have a deep foundational understanding of what they're doing. A lot of people kind of pit procedure and practice against conceptual understanding. And yet this teacher that you are talking about was able to weave those two together really well. And to provide an opportunity for students who have an understanding or have a have the opportunity now to practice sort of this, this procedure of factoring and make some real connections between those factor pairs and what that overall trinomial maybe it may have looked like.

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Joanie

Right. Yeah, exactly. And I think the, you know, I'm trying to put my finger on like, what is it about that sorting activity that took it to the higher level? And I think in general, sorting activities cause students to look more at structure and the reasoning behind what's happening mathematically rather than just attending to the procedure. And again, to kind of come back, I want to reemphasize what you said, that we often pit these conceptual understanding, you know, deep mathematical learning of concepts against procedures and answer getting. And oftentimes, especially in the past ten years, I'd say we've even seen some vilification of procedures and answer getting. And I don't know that that's because any math educator believes those are unimportant. But those aren't hard. That's that's been part of mathematics teaching and learning for centuries. Right? We already know how to do that really well. But knowing how to help students understand the mathematical concepts is harder to do. So I think that's in my opinion, that's why it's gotten the level of attention that it has. Not because it's more important than procedures, but because it's harder. And we we understand less about how to do it.

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 Start of Segment 2

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Joanie

Back to what makes the difference. I just pulled up the the standards for mathematical practice. Thinking back, could give me some language. And I think this sorting, really the sorting idea really forces students to go into math practice seven and math practice eight, where they're looking for and making use of structure and looking for and expressing regularity and repeated reasoning. And you can particularly picture that with that with my specific example, right, of factoring Trinomials when you're factoring, you're gonna notice the structure for, for instance, you know, factoring a and I said trinomial. So maybe this would wouldn't be the best example. But we often will see factoring with binomials as well. Right. A difference of squares or a difference of cubes. Factoring pattern. And recognizing oh hey, this is a difference of squares. And I know that there's a factor pattern that happens when I recognize that structure within a polynomial. So, getting at the ways of thinking about the mathematics is what took that from just fine. The factor pairs to really understand what's happening mathematically within these different, sort of expressions that are all grouped together on this page for some purpose.

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Curtis

Well, I had as you were talking about this, I was thinking about some homework that my nine year old was showing me this morning and I was thinking about, well, the reason he was showing it to me was because he had chosen not to do it last night, and now he was. 

00;11;43;19 - 00;11;53;14

Curtis / Joanie

It's due today. And I said, well, guess, be late.

You're because you're going to work on it this weekend. We're not nice. It's success one right.

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Curtis

That's that's you learning responsibility, pal. And so, so but I was looking at it and it struck me because we looked at the first problem and then the second problem. And in each case there was a, one of the, he was finding area of rectangles and the areas were, you know, it's a sort of arbitrary right numbers, except that I started looking at the numbers that were there, and it was a one digit number typically times a two digit number. And then I started thinking about how how would I multiply these or how how is it going to think about multiplying these? I doubt, I doubt that he's going to reason. The standard algorithm. Right. And sure enough the first problem was it was nine centimeters by 22cm. And I'm only telling this story because it relates exactly to your to your categorizing problem. So nine times 22, the very first thing he says to me is, well, nine times 11 is 99. Love it. And and so then I started looking at all of them, and the very next one had sort of a different problem with it. The factors of the two digit number provided the opportunity for him to exercise eight times eight. Right. So then he got this. So I was starting to look through these numbers like oh these could all be grouped. And I wondered if that was actually part of the way that they were. They weren't, they were all kind of random. But yeah. Yeah. At the end of the day what I was thinking about was all these little connections in there and, and I just ha, what an opportunity to take him out of the.So this weekend. Sorry about if you listen to this in 20 years yeah. We're going to go back to and we're going to explore what are all the different ways that those numbers in those areas are kind of related or interrelated as a project. 

00;14;01;21 - 00;14;15;13

Joanie

Yeah. And and what a great example. And again, this is maybe the blessing of children who have parents that are math teachers, workers. I wasn't allowed to say it that way because I, I know I'm talking about myself too. Here.

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End of Segment 2

Start of Segment 3

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Joanie

The other part of what I wanted to unpack with you in our conversation today, Curt, is, you know, we say this implement tasks that promote reasoning and problem solving, like, hold on, let's slow down with those words about, yeah, implementing or talking about that tasks. We're talking about that. But let's really talk about what reasoning and sense making and problem solving and what you're describing about what true it naturally saw, but what you're now going to capitalize on. That's just that was just such a beautiful example of reasoning and sense making. Right. Like the fact that true, it can say, okay, I don't know, nine times 22, but I do know nine times 11, and I know that nine times 11 is helpful in getting to nine times 22. Right? So with you asking the right questions and to generalize this to the classroom with a teacher who can uncover what student's current understanding is and build on it, that to me is what we're talking about for reasoning and sense making. I can't tell you how to reason because I can only tell you how I reason right? And my reasoning and your reasoning are not the same, right? Like my son Logan probably wouldn't have gone to I know what nine times 11 is, but he would have said, I know what nine times 20 is, and I know what nine times two is, right? So that's his reasoning in sentence making. Right. And and I think this is what makes the job of the teacher so challenging and so powerful is tapping into how students are thinking and then helping them use that to guide them to where they need to. They need to get to be able to understand. 

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Curtis

Yeah, yeah, I, I what you were saying is the implementation part, right? Yes. Like, yeah. The, the recognizing of the reasoning and the recognizing of the sense making and then the flexibility and really, frankly, my knowledge. Yeah. To be able to and I'm not saying my like me Curtis but my as a teacher teacher or knowledge of the task or of the mathematics involved, an inherent within that task is really where the implementation comes in. Yeah. And and you know, we talked about this yesterday and I don't know if this is a turn in the podcast or where we really wanted to go, but it struck me that, you know, reasoning and problem solving. I was just, about to. And for those of you who don't know, we record this on video chat. And I was just about to type Joni a note in there and think about, really, why did we why do we care about the conceptual understanding, and why do why do we want students to be able to sort quadratics into certain groups and the way that they factor? Or why do we why do I want to do it? To have an opportunity to recognize that one digit numbers, time, two digit numbers isn't just a procedure to be done right, but that there actually can be reasoning and sense making done about, the numbers inherent within that. Well, part of my reason is because I love mathematics. I find it intriguing and interesting, but I think a bigger reason is because there is reasoning and sense making and problem solving being taught. That goes well beyond the mathematics that a student needs to be able to do. Yeah, a student in most cases is probably not going to go out into the world and need to factor quadratics on a regular or even ever again in their life, right, exactly. But the processing and the reasoning and the sense making and the logic and the, the putting together and recognizing structure, all of the things that we've written into the standards for mathematical practice are the reason why we spend so much time teaching mathematics the way that we do. 

00;18;39;10 - 00;20;59;05

Joanie

Yeah, I, I would 100% agree with that. And I think what's coming up for me is you hit the nail on the head with the like all of the students who said to me through my many years as a high school math teacher, when am I ever going to use this right? And it's a very dissatisfying response to give a 16 year old to say, I'm teaching you the reason why I'm learning this is about how you think, and that's very dissatisfying to them. What I wish I could go back and say now is it's it from from okay, let me back up. Because kids are always saying, when am I ever going to use this? And my approach as an educator was I need to teach them what they need to be successful on the next assessment or the next measure of what they know. And then I would get frustrated when they didn't seem to remember that for the next topic. So math being cumulative, you need to know this because you need it for the next thing we're going to learn. Also very dissatisfying to a teenager, but I'm coming back around to reasoning and sense making because as an educator is often frustrated when you know, you give you work on a, on a, on a skill or an understanding of a concept, and then you give students a problem that looks a little different and they're kind of like, oh, well, now I don't know what to do because it doesn't look exactly like all the others. When we can build a foundation of reasoning and sense making and help students understand that that's a a wealth of a resource for them to personally tap into, then that allows for the learning and engagement of things that are different than we've ever seen before. And isn't that what life is? Isn't that the skill that we want them to be able to apply is how to learn and and how do I approach something when I don't know how to do it? Initially? Yeah. So it's it is about the long game, right? It's about it's not just about factoring trinomials, it's about recognizing that there is an underlying structure and that I can understand what that underlying structure is. So it can help me make sense of something I don't already know yet. 

00;20;59;18 - 00;23;58;20

Curtis

And, and then that what you just described is how we learn. I mean, I was sitting here thinking and contemplating, you know, do other subjects kind of run through this same sort of thing as well? Why am I reading Shakespeare, or why am I having to learn about grammar? Right. A five paragraph essay. Yeah. Figure out how to communicate. I mean, all of these things we're, we're we're really are trying to develop and teach and, and help students grow into contributing adults in, in their communities and, and the way that we do that, the way that they become folks that that are a part of society and contribute and help society progress and move forward is they learn about reasoning and sense making, and they learn how that applies to the different situations in their life. And and I, I don't know, this becomes a little bit philosophical. And at the risk of feeling a little bit philosophical, I think that's okay. I think that's what I think that's what education as a whole is, really what we're really trying to do, right, is, is trying to develop these, these young minds into mature minds that contribute. And I like that. I don't know if that's exactly where we thought this podcast was going to go, but when we think about the selection of tasks and we think about the importance of of trying to teach students the mathematics that we're trying to teach them, it isn't so that my son Truitt can multiply a one digit number ten or the two digit number quickly.

Right. He's got a calculator that can do that. He's got, you know, that those kinds of things can can be done and it isn't so that my high school student can look at a quadratic and tell me how to come up with the the zeros. Right. Certain engineering applications might find that important. And that's great. But the broad scale student isn't going to need to be able to do that. So why do I ask him to do that. Well I ask him to do that because of the things you listed off. Yeah. The the sense making the logic, the problem solving, the growing into putting together the parts of the puzzle that I do know and using those parts of the puzzle to try to help me figure out the parts I don't know. Right. And in I love that that's what we're doing, I love that that's how, we're talking about tasks. And in getting, you know, we said it at the beginning, not every task is going to be this huge, rich undertaking. And we need lots of opportunity for practice and and whatever. But even in those places for practice and routine and learning, the practice and the routine, those are pieces of the puzzle put in place so that I can learn a little bit more about reasoning and sense making.

00;23;58;22 - 00;24;30;15

Joanie

Yeah, I so I think where we've landed and again, like you said, I didn't know that this would be the punchline necessarily of the podcast, but the moment I'm having is when we promote reasoning and sense making with those tasks, how much we're empowering students. We are supporting them to be their own resource for learning and engaging with new knowledge and information in the world. And that's something for every teacher to stand up and be proud of.

00;24;33;10 - 00;24;51;15

Joanie

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!

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