SPEAKER_02 0:00

Our default is to go to tricks or gimmicks or, you know, acronyms or easy ways of remembering how to do something that is just about computation and answer getting. And it isn't rigorous, right? It doesn't build out this deep understanding. And I think without that, you're never going to get to that mathematical identity and agency that we strive for and ensuring that all students can see themselves as learners and doers of mathematics. Welcome to Room to Grow. I'm Curtis Brown. And I'm Joni Funderberg. We work together at Texas Instruments, and we're glad you're here.

SPEAKER_00 0:38

We're looking forward to continually improving our practice, and we understand that you are too.

SPEAKER_02 0:43

We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better. In this episode, Curtis and I talk about mathematical rigor. We start by defining what we mean by rigor and what we don't mean by rigor, and how it's more than just being hard. We discuss why rigor is important in the classroom, including how it contributes to equitable teaching. And then we give our thoughts about how teachers can be more intentional about teaching with rigor. So let's get growing.

SPEAKER_00 1:16

Well, Joni, it's a pleasure. We get a chance to uh hang out again and have a conversation today. Um and this time we don't get to be in person, but that's that's okay. We're um able to do this um across the internet, which is uh wonderful. So hope it's a good day there in Colorado. Um, today we're gonna be chatting uh a little bit about this idea of rigor. Um, and I don't know, maybe probably the best way for us to to kick this off is to decide what is, or at least have a discussion about what is rigor anyway.

SPEAKER_02 1:51

Yeah, it's a it's a it's a perfect place to start out, Curtis, because I'm sure as many of our listeners will relate to, people throw around this word rigor all the time. And and in different contexts and used by different people, you know, they're really intending it to mean different things. So I think it's a good place for us to start because as we get further into the conversation about, you know, why we think rigor is important and what it might look like to, you know, quote unquote do rigor in a math classroom, it's helpful to set that context about what it is we think that means. So, I mean, for me personally, I go back to, you know, seven years ago or so when I worked at Student Achievement Partners, and a lot of people know them by their website, achievethecore.org. And in their materials that came out to support educators in implementing uh the Common Core standards and other college and career ready standards, Rigor was one of what they called the shifts in math instruction that was designed to help educators understand how math instruction under these new standards should look different than it did prior. So Rigor was described by them as with like this image of a three-legged stool, that there's a balance between sort of three different aspects of teaching and learning mathematics. So the first leg of the stool is conceptual understanding. So really understanding what underlying mathematical ideas are happening within the math content that students are learning. The second is procedural fluency and skills, which is very, very important. You know, students need to be able to accurately do computation and know what procedures to embark on to solve a mathematical problem. And then the third leg of the stool being application, which is really referring to extending their learning, you know, beyond sort of a practice scenario and applying the mathematics they know to an authentic scenario or even just to a unique new scenario. So I think thinking about this blended approach and thinking about um mathematical understanding that is rigorous, meaning that it is robust and multifaceted. So that's just maybe one way to start the conversation. What do you what do you think about in terms of what rigor is, or even maybe what rigor isn't?

SPEAKER_00 4:09

Well, I think, yeah, I think one thing to think about with what rigor might not be, um, because I think it's a classic thing for us to come into um, you know, thinking of rigorous as uh more difficult or setting up and making something that is harder. And I think in some regard that can be that can be partially true, right? That when a task or or a challenge or an activity um is rigorous, often um there's more to it than maybe a simple task might require. Right. So there's there's something uh to be said about, well, yeah, it is potentially more difficult, but not just more difficult. It isn't that we just make something challenging or uh make a task include a whole bunch more things or you know, things of that nature. I think that's that's not um that's not it totally. There's there's a potential for it to be at least not simple, right? Right. But um when I was thinking about this earlier and I was reading and and um trying to think about it, I the the term rigor kind of comes up in like geometric things, right? Right, you do these rigorous geometric proofs, right? And and what that what that means in geometry, you know, the the concept of of it, you know, there's really no there's no holes, right? There's no holes in my proof. My proof is is unquestionable. And that that brought to my mind um complete, uh well-rounded, right? And so when we think of what is rigor and with regards to a task, and you mentioned the three legs of the stool, right? Um, just the the idea of of conceptual understanding procedure and application kind of interwoven between the three of them. That also to me feels very stable, very complete. Um, I like that. And it and complete in the way that something, that something works. And I related that to the way we think about uh geometry and geometric proofs. And so that's kind of how I've been thinking about this this idea of rigor and how it applies to tasks that we might give to students, or how it might even apply to an activity or a lesson uh that we do with students, or even if we go in and and we think about um, you know, the way that a student actually is able to use the mathematics that they know and they understand, right? They they could have um a more complete understanding of a particular topic area or uh concept in the mathematics. So those are things that all kind of came to my mind when we started thinking about what is this rigor anyway.

SPEAKER_02 7:06

I I love that. I love that you brought up the part about hard, right? Or challenging or difficult. Like there is a piece of challenging in rigor. Like I think that's sort of a natural part, but it's not just hard for hard's sake, right? Like let's pick the most challenging numbers to have students engage with so that it feels, you know, some kind of way. And it kind of reminds me of previous conversations that we've had around productive struggle, you know, and like there's this level of challenge that's appropriate that causes us to grow and deepen our understanding. Um, but then there can be a place where struggle becomes unproductive and just frustrating. And we know that that leads to students shutting down, not engaging more deeply in the learning. And just one more visual that some of our listeners might also be familiar with, Kurt, one of the other resources you and I quote from all the time is the uh adding it up publication from the National Research Council. And I think they have a really nice um, and I think it's about mathematical competency, is how they label it, but it's a rope analogy or a rope visual, where, you know, a rope is not a single strand, a rope is um multiple strands interwoven together to make it strong. And they bring in productive struggle alongside or productive persistence, I guess is what they call it, alongside procedural fluency, conceptual understanding and application. And then their fifth strand is actually the strand of rigor. So I I really like this rope strand of thinking about rigor as being a part of it, but also really all the full comprehensiveness of the rope, I think is also rigor. But one of the things that's uh in that definition that maybe we haven't brought up yet is this idea around effective communication and students being able to clarify, be really clear and precise as they're talking about and communicating the mathematics that they have, you know, done and understand. So it's and I I think this is a nice way of rounding that out, right? Like it is this sort of higher level of interaction, right? When when a student is just learning a new mathematical idea, I think about, you know, my fifth, my fifth grade nephew, your kids that we talk about all the time. And they'll, you know, kids will use words that aren't mathematically accurate, like, I'm gonna times these two numbers. And it's like, okay, like timesing is not something that you do in math. And as they get more proficient and as their their the rigor of their understanding and their flexibility with that content develops, they know that actually I'm gonna multiply these two terms or I'm looking for a product. Like they start to get more concise and more precise with their language. So I think that language piece is a nice, um, a nice connection to also how we define rigor.

SPEAKER_00 10:03

Yeah, I think so. I think so for sure. Yeah, that that idea of being able to talk to uh the mathematics, right? And be able to explain my thought process with cor correct and accurate and precise mathematical terminology, right?

SPEAKER_02 10:22

Exactly, exactly. And that connects to right the mathematical practice number six of attending to precision. Like that's really about that um conciseness of language and accuracy and precision of language. So okay, so with kind of this understanding of what we're talking about for the rest of this episode as we talk about rigor, let's shift now to talking about why is it important? Like why are we dedicating a podcast episode to talking about rigor? What does this, what does that have to do with with teaching and learning?

SPEAKER_00 10:57

Well, I think it goes back a little bit to the things we've talked about in the in previous podcast episodes around, you know, conceptual understanding and procedures and how those two work together for students having a foundational uh abilities, right? This uh idea of rigor, I think goes um, takes it another step further, right, with the application piece of the of the puzzle there fitting in together. Um, in my opinion, or at least in the way that I feel about this, this idea of rigor, why it's so important, gets students to be able to really internalize everything. We've talked about conceptual understanding and its importance in terms of the introduction uh of concepts and that we don't just jump into procedures, um, or at least that's my opinion, right? That we that we talked about um kind of getting at what is what is really going on here. And then we can look for efficiency um and develop efficiency through those procedures. But then we can take this a s for a step further and and apply it to things um and weave in that that idea of of application and purpose and seeing uh seeing this mathematical concept not just as an abstract mathematical idea, but really as an application to describe or um work within physical constraints or things that apply to the world that help us understand the way the world works in that way, right? So like I feel like that that it's super important because not only is it application, but it also brings everything together. And then because of the bringing everything together, it makes it stick. It makes it a part of, it makes it a part of my memory. And I think that's super important.

SPEAKER_02 12:51

Yeah. I think too, the the stickiness of it is has to do with really deeply understanding, right? Like when we go back to all of those components that we talked about in our definition a couple of minutes ago, uh if I really do have conceptual understanding of the underlying mathematics and I do have procedural fluency where I'm I'm flexible and I can manipulate and manage the numbers or the algebra really well, and then the application, which I know as a classroom teacher, that's so important. And so many of the educators that I know struggle when students don't remember something that we taught them last week or three months ago or in you know a previous grader course. And that can be so frustrating. And I think really teaching with rigor front of mind and being intentional about ensuring that teaching and learning is rigorous helps address all of those things. Um, but I'm also thinking a little bit about the connection to the ideas of mathematical identity and agency. So, you know, in order to, in order to be so confident and clear in my understanding of mathematics that I can apply it to a completely unique new situation that, you know, the teacher has never shown me how to do this kind of problem before. This is a this is a mathematical situation that I've never encountered before. But I have faith in my own ability to draw from what I know and apply it to this new scenario in a way that will allow me to make progress, right? And I'm I'm hearing productive struggle coming into this as we're talking and all of that. But it's it's about my own ability to think of myself as efficacious as a mathematician. Like I create an identity around I can, um, I can do hard math, you know. So I really think of this as an equity, uh, an opportunity, an access to math kind of concept as well. And, you know, in my experience in the classroom and working at the district level, I was often frustrated and just disheartened by a lot of our intervention approaches or ways that we manage students who struggled in mathematics. And I know we've mentioned this before in previous episodes, but I think it connects to this idea of rigor too, because our default is to go to tricks or gimmicks or, you know, acronyms or easy ways of remembering how to do something that is just about computation and answer getting. And it isn't rigorous, right? It doesn't build out this deep understanding. And I think without that, you're never going to get to that mathematical identity and agency that we strive for and ensuring that all students can see themselves as learners and doers of mathematics.

SPEAKER_00 15:45

I mean, what you're bringing up is such an important piece to this whole puzzle. Um, the idea that rigor is, in fact, for all students. I think part of that might stem, and you can you can nod your head, confirm that, or shake your head, whatever. But the the one of the reasons why we don't see it happen is because we continue to have this thought in the back of our minds that rigor is more difficult. And therefore, if my students struggling, why would I give them something that's more difficult? But what you brought up, but what you brought up is so important that, you know, I mean, we've we've already kind of established that, you know, we believe that it that's the connectedness that makes it sticky. And if it's the connectedness that makes it sticky, then that has to be how we're working with all of our students, right? But then you also brought up the the agency and identity piece of uh and the confidence that that I can go forth and do these things. And so exposure to rigorous material is so crucial for all of our students to have access to the that kind of material, to have the access to that kind of uh teaching and learning um is super important for them. Because if we're just stripping it down to only the procedures, we already know that on only the procedures becomes a memory, a rote memorization tool, right? Right. And we just memorize these things. And the thing about memorizing things, um, I don't know about you, but I've memorized some things in the past and I've had to say them or you know, give a speech or you know, sing a song or whatever. And yeah, during the process and during the time, I was able to memorize it and I was able to regurgitate it at the appropriate time in the appropriate manner, whether it be so I used to sing, I don't know if you know this, but I used to sing at a barbershop quartet. All right. I didn't know that. I did, yeah. And so we would sing songs, we would memorize, you know, we would do all these things. But I could tell you right now that if my group of guys got together and we tried to sing a song, there's no way because we haven't been practicing, we haven't been doing these things, right? So it was memorized. It was it was totally memorized. But if in in in the same way, I I was, you know, I think of I was just thinking about driving a car. Like you learn how the car, how the how the gas pedal works, how the brake works, right? And and you do those things without really thinking about them, right, most of the time. But that's because you're doing it all of the time. You're doing it, doing it, doing it, doing it. That's what we get to when we're doing all this procedure memorization type stuff. Sure, we can get students to where in the time, at that time, maybe they can do this, but it hasn't become something that they go on and have a deep understanding, a deep foundational understanding of the way that this works and they can apply it to something else because all they've done is just procedure, procedure, procedure. Right. Um, and and it hasn't stuck. And so the importance, I I can you know, I got on a little bit of a tangent there, but I I feel like the importance of rigor for all students um it is so big because if we only do the small things, if we only do the memorization and the procedure piece, which is an important piece, don't don't make me. We're not saying we talk a lot about right now, we talk a lot about all of these other things being so important and we lose sight of procedure sometimes and we go, oh my gosh, it's not you know, procedure, uh procedure's bad. No, procedure's not bad. It's a part of the puzzle, it's an important leg. Right, it's an important leg of that three-legged stool. It's an important leg of the stool for sure. But the but there are two other legs that also need to be uh a part of everything. And if we're not doing and working with conceptual understanding and then applying it in a rigorous manner such that we kind of have this complete peace, we neglect students and we don't set them up for success.

SPEAKER_02 19:53

I think that's so powerful. I I sorry, I just want to add like the the way you brought up initially that how when we sort of default to that thought of rigor is hard. And then we think about students who are struggling to learn mathematics, and you know, that idea of struggle means they're finding something hard. It would not be our default to go to something more rigorous because, oh, it's already hard. We don't want to make it harder. But I think when we step back and think about that broader perspective of what rigor is, for students who struggle, rigor might just be exactly what they need, right? To build that deeper level of understanding and move away from the reliance on procedure. And again, not to say procedure and correct answers are not important, but sometimes there, if there's an over-emphasis on that, sort of zooming back out away from procedure and building some of these other ideas around, you know, what does this really mean? And how does this connect to other mathematics that we know, or how does this connect to what we understand in the world might just be the solution to moving kids through that. So I just kind of wanted to stop and take that breath for a minute because I think that was really profound.

SPEAKER_00 21:03

No, I think that's I that's a really important piece of the puzzle is that stepping back and realizing that it's possible. It's possible that that rigor might be what those students really need. So here's an interesting question, Joni. Um, and it came up while we were kind of setting this up and and um you know, when we were trying to think about how this podcast might go. Uh, this concept came up in our conversation. Okay, it's really great to have um these high level conversations about, you know, rigor's important, and here's this three legged stool and whatever. But practically, how do I do rigor quotes uh in my in my classroom? Like how is this? Even possible in my classroom.

SPEAKER_01 22:02

Well, I hate to be my a Joni broken record, but I feel like I say this all the time in every episode. Like, how do we implement this? Well, you have to do really good lesson planning.

SPEAKER_02 22:14

But I but I really do think that that is so important here of like the teacher's intentionality around, you know, being sure that instruction is this multifaceted approach to understanding and not just an emphasis on how to solve a problem or how to get to an answer, but really asking the kinds of questions that help students make connections to previous knowledge, to things that they've already learned. And then asking the right kinds of questions or creating the right kinds of experiences in instruction that allow students to make sense of, to come up with generalizations about, to um, you know, really be able to put the mathematics that they're learning into the context of their own understanding. To really, you know, the way we've said it in previous episodes, Curtis, is for them to really own the learning, right? For it to be, you know, something that they internalize as this matters to me and this is something that I care about. And I think, you know, there's this certainly this piece of the chicken or the egg around conceptual understanding and procedural fluency and how do you, you know, which should come first. And I think this is maybe the best argument for sort of that cyclical reiterative approach through both of those. Like I think there has to be some procedure that comes along. And then um, and then there also has to be conceptual understanding that's that's thrown in. So um I just want to share again old standby for our regular listeners, you know, thinking about the fifth graders that I tutor. And we were spending the end of the school year talking about division of fractions or dividing a unit fraction by a whole number. And I got them, you know, we we were drawing examples to show, you know, what does it mean to say one third divided by three? Like what would that look like with a bar model, for instance? And we did so many of the examples that they were seeing the pattern between the numbers, right? Like, well, one third divided by three, that's one ninth. Well, there's a three here in the bottom of the one third, and there's a three here that I'm dividing by, and three times three is nine. And they could really easily see that pattern. But I just kept asking, well, why is that, why is it doing that? Why is multiplying those two numbers connected to the answer? And they didn't get it. You know, we we talked about it for a couple of different sessions, and they saw the pattern and they recognized that it was there, but they could never explain why. But continuing to ask that and continuing to provide them with examples to sort of uncover what's happening mathematically in a way that could make sense to them. You know, I wasn't worried about, okay, in this, you know, 60-minute or 45-minute tutoring session, they're not going to be able to answer why is that happening at the end. But I've planted that question in their heads. And, you know, when we start up tutoring again at the beginning of sixth grade, like we'll come back to that. And as they've learned other things around with division of fractions with other kinds of values, you know, keep asking those questions. So back to, I'm going off on a tangent here too. How do we do rigor in the classroom? I think it's just with intent. And I think um, you know, the other punchline of the story I was just trying to tell is it it's gonna take time. It's not something that you're gonna say, oh, this is um, you know, one of my learning goals for today is to teach this concept with rigor and kids will have this deep level of understanding by the time they leave. You know, that doesn't learning doesn't happen like that.

SPEAKER_00 25:45

No, that's that's true. And but I think you're making a very uh a pretty important point here. And so I'll just um follow, follow up your point by maybe even reiterating it that planning is so important. Because rich richness and um I like the I like the uh the analogy of richness there. Just you think about chocolate and big deep taste and whatever, right? Like that just, I mean that's what the image that draws up in my mind, right? Like all of this is it's so full. And that's why completeness, when I was talking about a geometric proof earlier, like this concept of completeness, that's what we're talking about. But this idea doesn't happen by accident, right? I can't just I can't just show up one day and and hey, we're gonna have this rigorous task and you guys are gonna go and and do this, because there is a tremendous amount of uh teacher involvement um that happens to actually drive this uh experience. Because I think a a rigorous task, inappropriately administered, becomes a really difficult task. Does that kind of make sense? So the idea that if I if I take a rigorous task and I just kind of plop it in front of my students and say, okay, um, without my own self having uh had some level of preparation and thought and prep and thinking about, you know, what are the responses that I'm going to have uh come in, what are my responses going to be to those responses? How might I uh sort of gather my students' um work and and their draft work, if you will, um, and sort of build it together so that we can develop what might become a model response to this task. And if I haven't done all of that stuff in the pre-work, if I haven't really thought about this, um then I then I do kind of um run the risk of having this task or having this experience become a really difficult one and one that doesn't benefit, especially my students who might be struggling, right? They're the ones who are going to ultimately end up becoming frustrated and that kind of thing. Where if I've got the right level of preparation and I can know exactly what kind of question uh my my student needs to be asked in order to prompt them to the next sort of uh thought process and and discovery. Um if I can remember our our uh learning pit conversation, right? If I can help to show them the rope, turn them to the rope, um and and not like you know, help them climb up the rope, but just point them to the rope. If I can, if I have prepared well enough, then I can do that. Um but if I'm if I'm not prepared, if I haven't really designed this task, things can get off the rails in a hurry.

SPEAKER_02 28:50

Yeah, I think that's so important because it's not just about, oh, I should do the task first as a teacher so I know the right answer, right? Like it's also about, and I think this is where the importance of planning with other teachers is so is such an important aspect of that too. And I'm thinking especially about our early career teachers and you know, tied back to the ideas around struggling students and rigorous for everyone, you know, it's not just about when kids get to AP, they should be engaging with rigorous mathematics. They should be engaging with math, rigorous mathematics all along the way. And too often those lower level courses, and I'm speaking from a high school perspective, but certainly, you know, the courses that are designed for the students who aren't typically successful in mathematics are are oftentimes taught by educators who don't have a strong mathematics background. And in order to, you know, fit this teach within the scope of rigor that we've been talking about, you have to really understand the mathematics yourself. Because understanding what obstacles students are gonna encounter as they're engaging in this rich task allows you as a teacher to decide, okay, which obstacles are gonna veer them so far off course that it's gonna create unproductive struggle or it's gonna take them to some mathematics that isn't my intent today. And which obstacles do I actually want them to hit head on so that they can work through and engage with and use that productive struggle to build this robust understanding of the mathematics I'm trying to get at. So it's not just about knowing what math the kids are gonna do, but it's really knowing and being able to be intentional about where do I want them to, you know, really get into the muck? And where do and where do, you know, oh gosh, if they're going this way, that's not gonna lead to something productive in terms of today's goals. So I'm gonna steer them back. What's the question I'm gonna ask if they're not even seeing the obstacle? You know, if they're if they're feeling like, oh, this is easy breezy and um, you know, not encountering something that's gonna lead to that development of understanding. So I think, you know, planning can just be a really trite response. Oh, teachers just have to plan and be intentional, but really that's it, it's in depth and it's it's important and it's uh it's time consuming and it's collaborative. Um, and it can make or break the students' learning experiences, I think.

SPEAKER_00 31:24

And Joni, you brought up a really interesting point uh about planning with others and talking to others, um, and bringing up the idea that, you know, oftentimes um we as teachers who are, you know, if we're we're teaching uh a course that that um you know maybe we maybe we don't have um the background or or whatever, the importance of being able to rely on others. Because, you know, I I don't have I I probably can't go out and just uh all of a sudden fix or you know, respond to, you know, well, I don't really know this one super, super deep. Well, that's why we have these other folks around us, right? That's why that team planning is such a such an important thing. I think back to uh Kathleen Morel and um some of the other teachers that I used to work with. Uh Kathleen, if you're listening, is thank you so much, right, for the uh amount of time that she spent with me talking about uh the content that she was teaching. Not that I hadn't learned it, not that I hadn't uh seen it, but just the importance of the way that I set my students up for the classes that she was going to be teaching and having them in. Um, and me being uh aware of and really deeply understanding the concepts. And I think that it's the concepts. I didn't have to be fluid in all the procedures of the things that were going to be going on in her AP calculus courses and the other, you know, other courses that she was teaching. If I could be fluent in the concepts and in the connections between what I was doing and how I was setting things up here, um, and the connections between um within and among the concepts that I was teaching in my course and how that really set them up for the things going on uh in the future, that was really important. So I relied on and I leaned on uh others around me. And I think that's a big encouragement, right? Of this team planning and hey, I'm gonna be doing this thing next week. And uh, you know, what are your thoughts on this? Um, I'm thinking about this activity. What do you think about uh doing this? Or hey, you know, go down the hallway and and not being uh embarrassed. I mean, shoot, I'm telling our podcast listeners out there that I had to do this, right? That I'm struggling. So I I totally think it's you know, just being willing to go and ask the person next door. Um or if you don't have a person next door, find a person next door uh that you can talk to via email or you know, go to a conference and meet someone and exchange emails and and um get connected.

SPEAKER_02 34:16

Yeah, get involved in your local NCTM affiliate or or go out on the my NCTM um discussion boards. There's some great conversation and great learning that happens there that you know you can do from the privacy of your own classroom. But yeah, that's exactly 100% echo that. And I know Curtis, when I when I was teaching at my best, uh it was because I was planning with other teachers and helping, letting them help bring me along.

SPEAKER_00 34:43

That's exactly it. Well, Joni, this is uh I think this has been a pretty productive uh conversation for both of us. Um, I have some things to go back and to think about uh whenever I'm thinking about tasks and planning them. And um, so I'm really excited to um go forth and put this into action. Um, look forward to chatting with you again next time, huh? Thanks, Kurt.

SPEAKER_02 35:09

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.