Latest / The Authority Podcast — Expert Insights and Fresh Ideas for Education Leaders / Universal Design for Learning in Mathematics Instruction with Katie Novak and Ashley Marlow
Transcript
- Ross Romano: [00:00:00] Welcome in, everyone. You are listening to the Authority Podcast here on the Be Podcast Network. Thanks, as always, for being with us talking about some really timely topics today. At least timely in my world, but I think timely in a lot of your worlds as well. And it's all around mathematics, right? So my guests today are Katie Novak. Katie is an internationally renowned education consultant, author, graduate instructor at the University of Pennsylvania and a former assistant superintendent of schools in Massachusetts. And her co author, Ashley Marlow, who is an elementary math coach, consultant, and the director of Operations for All Learners Network. Katie and Ashley have a new book out called Universal Design for Learning in Mathematics Instruction K [00:01:00] 5. Welcome. Katie Novak: Thank you so much for having us. Ross Romano: You're used to, you're used to your longer bios, but I decided today, you know what, we'll let Katie Novak: No, I love Ross Romano: that for themselves. And we are going to, we're going to jump right into it. Right. Would love to start out with Ashley. Some of the context around this book and why this book is important specifically tying into math and you kind of write in the introduction to the book that math is one of the areas with the biggest hurdles to access and inclusion. Could sort of fill in some of the details why that is or how that is? Ashley Marlow: yeah. So my background, I was a classroom teacher turned math specialist math coach and what I was experiencing in the field alongside my colleagues with this You know, my, my educational experience led me to this path of a much deeper understanding of mathematics from the content perspective, but also the pedagogical perspective and what I was seeing in the field of among my colleagues was [00:02:00] this huge challenge in knowing what to do when their students show up. Not understanding mathematics, and that trajectory is a really different experience for how folks feel about teaching literacy. So when we think about our students experiences when they come into our classrooms, our teachers feel more confident. are more prepared to meet the variability of our students that show up in our classrooms when we're talking about a literacy perspective. But in mathematics, teachers not only didn't really learn math in a way that allowed them to see the wide variety of entry points into mathematics, but then a big challenge is that the pre service training programs provide much less focus on mathematics. We see that actually happening. This is not just an elementary issue. We see this pre K 12 because then when we get into the secondary grade range, they're so focused in the content that they teach the [00:03:00] courses that they teach, that they're not getting the experience of understanding the mathematics that built up to that. What were the prerequisite mathematics? So when we talk about this being a huge barrier, we're really talking about Teachers don't have access to the tools that they need to be able to do this well from a training perspective and from a resources perspective, this one size fits all approach, looking at our core programs, a program cannot be written for the human beings that show up in your classroom. So when that is the only tool that you have in your toolbox, it really makes it difficult for teachers to meet the needs of the students who are in front of them. And we just See this being a much greater issue in the field of mathematics when compared to the other content strands? Ross Romano: And I follow up with what I mean, perhaps it's a dumb question, but for lack of a better word, it seems weird to me that if like reading and math are kind of the two [00:04:00] most tested subjects kind of considered of the core curriculum, sort of the two predominant, Subjects and areas of knowledge that math would be under emphasized in kind of pre service learning. Ashley Marlow: Yeah. I think there's a lot of reasons to that. Number one being people assume that the program is going to be the fix, right? That's gonna be enough. There was also a large body of evidence supporting. The fact that reading is a student's ability to read would be the greatest predictor of high school matriculation, college and career readiness. We just weren't talking about mathematics in the field in the same way. We were very focused on the benefits of literacy. And then, as we talked about in the book, the Algebra Project in the early 90s. really illuminated the fact that mathematics is actually the greatest gatekeeper to college and career readiness, high school matriculation. When we look at Algebra 1, in [00:05:00] most states, it's a required course for graduation. But what happens when you get to Algebra 1, and you are not prepared to access the material in Algebra 1, meaning you have several years built up of misunderstandings, that's when we often start seeing students either dropping out of school altogether or really struggling to graduate from high school. And as we talked about in the book, This is a huge equity issue because the students who we are talking about who are predominantly missing out on the opportunities that Algebra 1 affords are our students who hold more marginalized identities. They are the students that we are constantly talking about. Students with IEPs are black and brown students, multilingual learners. So when we talk When we're wrapping our heads around the fact that not only is this a, it should be equal in the conversation of literacy, but actually we now know it's, it has an even [00:06:00] bigger impact, right? And so then when we think about this, the jobs that don't exist yet today, they are going to be in the STEM field. The jobs that we are, we want our students to enter into to be problem solvers, to notice the problems in the world and think about how to solve them, they will likely be in the STEM field. And so, for me, this book was really important because I believe that teachers, when they have the tools that they need to do better, they will. And I needed to be part of a project that was illuminating, not just the challenges for teachers, but how this is really creating unhelpful barriers for our students that have major impacts on their lives and for the generation to come. (ad here) Ross Romano: Katie, within a lot of what Ashley was just describing, I think surrounds like a call to action that's within this book, which is building on, as she mentioned, the algebra project [00:07:00] and the conception of math and rigorous access to rigorous math instruction as a civil right, not just a good thing for schools and that Kind of target of algebra and algebraic knowledge as that gatekeeper to future success. You know, I think that sort of communicates a lot of the motivation and urgency around tuning into this and ensuring that, creating math, learning that all students have access to. I guess the question within that is, who needs to hear that call to action and and begin to contribute to building you know, learning environments where by this can happen. Katie Novak: So I can speak a little bit just about the fact that what we're doing doesn't work. And I don't know why we continue to use the same practices that we've been using [00:08:00] historically when it's not making a desired impact. So the 2024 Nations report card was recently released. And in it, when you're looking at data, 8th grade proficiency only 24 percent of 8th graders are proficient in 8th grade mathematics. 24, but not even a quarter. And so the reason that we're so focused on elementary mathematics is like, what is the foundation to be able to access, right? That kind of algebra hurdle that happens basically in 8th or 9th grade. But really interestingly, That was an increase from the previous so like we cannot celebrate that three quarters of American students do not have basic proficiency in mathematics. And a lot of the times we're talking about literacy, but the literacy scores are only 30 percent proficient. So even though there is a little bit of an increase proficiency, what we know is that there's. So much focus on the science of reading that like there has been [00:09:00] an international reckoning that the way that we have been teaching students to read is not effective. And that is absolutely starting to scale that we're seeing literacy instruction change. And I would fully expect with that investment for the literacy scores to continue to increase ultimately until we're at a hundred percent, we have work to do obviously. But when we're thinking about. coming from a 24 percent without a science of mathematics boon coming internationally you're really relying sometimes on individuals within a system who are championing doing things differently, but that requires that they even know it's possible and what it looks like. And of course, when we talk about increasing the outcomes of students we're generally looking at four key levers the first is really effective instructional practices. What does excellent teaching look like and how can that be measured in the impact on the students that we serve? As much as teachers are working [00:10:00] so hard, they're doing the best they can with what they have. Most people are doing exactly what they were taught to do. And we're just not seeing that impact on students. So. What are the effective instructional practices of today? And the other is high quality instructional materials, but that doesn't have to equate to scripted program and teacher holding book reading from script, right? And so how do we find this really beautiful balance between really effective instructional practices like universal design for learning, like differentiated instruction, like blended learning models and leverage really high quality instructional materials so that we can better meet the needs of kids who are serving. And so when you're talking about the four levers of what really increases student achievement, it's evidence based effective teaching practices. It's high quality instructional materials. It's data based decision making, which is. How do we know what is effective so we can keep doing it? And then how do we solve the problems [00:11:00] of Student disengagement and students who are not making progress and then ultimately the last is this tiered supports, recognizing that some students may need additional support, but I cannot stress enough that it's supplement not supplant if I had a bumper sticker That's what it would say because I think a lot of the times we look at these scores and we say oh my goodness 75 percent of students are not proficient at mathematics. We need more interventionists. We need more special programs. We need more small group instruction. But we really have to focus on what is first best instruction and what is the evidence base. Effective practices that we have to use. What are the really, truly high quality instructional materials that we have to use? And then how are we going to consistently use data to increase the outcomes of students before we're thinking about how we're going to intervene our way out. And Ashley, you talk a lot about in the book, just how you know, you've worked with schools that have lots of different [00:12:00] programs and the program is not the greatest indicator of success, but the Ashley Marlow: It never is. Katie Novak: that is used with the program. Ashley Marlow: using the same ones and they're all having different impact. Right? Because we know that the program is not the thing that's going to be the change maker, and the only piece that I would add to everything that Katie just said is you cannot design high quality first instruction and a tiered layered intervention support without understanding the math that you're teaching, without understanding the human beings that are in front of you. So all of those pieces that Katie is describing as the change makers that we know work in education. They work for students. The problem is teachers are not being provided the tools that they need yet to understand the math at the grade band that they teach and the mathematics that came before that. And then how do I determine what is the most impactful mathematics to [00:13:00] intervene that to intervene on? What are those skills that they need to access more of that? Really beautifully universally designed first instruction. We've gotta put all of those pieces together. And that's part of why Katie and I organized the book the way that we did. It's a nice blend between here's some pedagogical decisions that you can make, here are some things that you can do to design your learning environment. To provide more ramps for your students to access grade level learning. And then here's the math that you need to understand. And it might feel different to experience that mathematics in a way that is not how you were taught. Which was one size fits all. Let's just follow along with the teacher and hope that we remember what to do next year. Ross Romano: Supplement, not supplant is a good concept. I don't think it would be a great bumper sticker because I think people would be reading it, trying to figure out what does this mean? And then they would end up bumping into your car while they're like, Katie Novak: You know what, you're right. You're Ross Romano: I would Katie Novak: The bumper sticker is pet more [00:14:00] puppies and the tagline for my PD is supplement not supplant. Ross Romano: Yeah, bumper stickers need to require minimal thought. Katie Novak: You know what? Touche. Ross Romano: to a lot of what you just said. Like, I think maybe a great place to go with that is the discussion around professional learning. You know, Katie, you referenced the results, right? The nation's report card kind of introduce the 2022 results are part of the introduction of the book. We're recording this a week or 2 after the 2024 results came out. A lot of the trends held consistent. Right? Some widening of. Achievement gaps between the highest performing lowest performing students. The math results actually relatively good compared to the reading, but but none of them where we would want them to be. And 1 of the takeaways. I would have from them is like, not that we don't know what works, but that we don't do what works and we don't do it consistently enough and everywhere, right? Because there's [00:15:00] clear signs in places where there is fidelity of implementation around. A lot of the things you've mentioned, high quality instructional materials, right? Research based data informed practices that there, there are positive signs, but it's not happening consistently enough because of perhaps similarly to when we've had. Some previous conversations about UDL, like it's only happening some places right when it could benefit all students and there's opportunities to scale some of this. So I think the professional learning would be a great place to say, okay, how do we tie this together? All right. We know that largely based on what states are requiring, what districts are doing, right? There's investment in The program in the materials that are meet the standards of quality, right? That it's but on their own, it's how they're taught how consistently they're taught. You know, what? What is [00:16:00] how does the teacher understand what it's supposed to look like? Do they have access to the data to be able to make informed decisions about what students need and what's going to move them forward? And. Having high quality professional learning would be one driver of that to ensure that that understanding is there and that that practice is present. I guess, Ashley, I'll start with you just on what kind of decisions. Can and should be made around professional learning to start bridging the gap that you referenced with not as much emphasis in the pre service years on math instruction. What we're clearly seeing with achievement results that we want to improve and. That certain variables aren't that variable right now, as far as the programs, the curricula, all of that, that there's a solid understanding of what those need to be but teaching and learning happens in the classroom, and the [00:17:00] people in the classrooms are teachers and students. And so how do we make sure that those people are set up for success? Ashley Marlow: Yeah. So when I'm working in schools and I'm talking with system leaders, we know that we can't come in and just try everything, right? Teachers are overwhelmed right now. Teachers need to feel success. And Katie and I talked about this a lot as we were writing the book. How do we write this in a way where teachers are feeling Positive about what they are doing, what about their own math thinking about their own way of teaching math about their own math understanding. Like that was a big part of our book intentionally because we have to think about how we're going to shift teachers mindsets if they're going to be able to shift their practice. And then that's going to trickle down to their students, right? So the first thing I think about is what is teacher's content knowledge? I know I'm harping on that a lot tonight, but how are we providing instruction on the actual mathematics? How are we supporting teachers in understanding fractions? Because [00:18:00] keep change flip is not going to fly anymore. Keep change flip does not carry on and help students to understand how fractions work when they are applying it in a proportional situation or when they're solving it in an algebraic expression. You can no longer keep change flip. Right when we're talking about subtraction, just cross off and carry the one that does not carry out over building and breaking apart numbers. We need to be able to do that starting from pre K within five all the way through until they get to algebra. They've got to be able to build and break apart numbers, compose and decompose. So what are some of those key mathematical ideas that are enduring over time that have a major impact on students? Mathematical understanding. Make sure right. Teachers understand that first, right? So there's the content knowledge professional development. That could be the right starting point. I recommend that when folks are adopting a new program, teachers are getting used to just how their program is set up. [00:19:00] That's a whole year long project in itself. How do I understand where my materials are? How do I determine where to find the high quality materials? And then my precious professional development time is skilling up teacher content knowledge. The other piece is professional development around pedagogy. How do I get kids talking to each other? How do I sort and sequence kids work? How do I support the integration of concrete representational and abstract models? Do I understand how those models Are embedded throughout my main lessons and how do I strengthen those for students through my targeted interventions? Those would be the three kind of key areas. I would think about when schools are thinking about professional development Am I going to invest in content knowledge? Am I going to invest in pedagogy? And then that final piece is how do these pieces fit together with my high quality resources? I believe that you can start in any one of those lanes. There's not one right And I know this [00:20:00] is a really important starting point in my experience, but I know starting with all three at the same time doesn't work. And I know that spending an entire year and a lot of money just flipping through the book is also not going to do it either. So how do we balance skilling teachers up in the areas that they need, particularly around math content, knowledge and instructional like what actually works in teaching mathematics. And then we support that with those high quality resources that you are investing a lot of money in. Ross Romano: Yeah. So, Katie, I guess, I mean, I don't know, maybe you can disagree with this. I don't even know if this is right, but we can have rigorous math instruction, but if it's one size fits all, right, then it's not accessible. And then we're, we have a problem, right? Maybe you'll say it can't actually be rigorous if it's not accessible. I don't know. But for lack let's say the teacher has the content knowledge, Knows how to deliver [00:21:00] instruction based on the high quality materials, right? According to practices where we wouldn't say, well, this isn't really math or this is low quality math. But. There's that hurdle to overcome of why is this not still more accessible to more learners? What are the struggles around that? Why is it not happening? How do we make it happen? So can you layer that in for us a bit about, like why that continues to be a struggle and then bringing in some of the solution to that? Katie Novak: Sure. So, what I will say is as, as curriculum organizations, companies create new content, it is becoming more and more flexible and accessible. Just by nature of digital technology, by nature of the math. Practices for common core, which require students know multiple ways to solve a problem. They're critiquing the problem solving of other people. And so what I will say is that if [00:22:00] you are getting high quality instructional resources, they are often well designed for many students. But the evidence base is not targeting particular groups of students. It's really looking at students at large. And so you might say when we use these materials. You know, with 1200 students in a number of different districts off the Midwest, we saw that it had a really positive impact on student growth in math. That doesn't mean that it had a positive impact on every single learner. And so there's not a lot of isolation of was it, what's the evidence base for students who have significant intellectual support needs? Or what is the evidence base for students who are emerging multilingual? We don't have that information because they're not. I'm not studying that in isolation all of the time, but I do think that it provides a really good foundation if we know how to use it. And so one of the things I always say is I work with a lot of teachers who really do [00:23:00] feel like they're supposed to just read it exactly as it's designed. And what I always say is, like, if the program was that good. That you could just read it, then you don't need teachers, right? If the program was that good, then all you need to do is get anyone to come into that room and read the script, right? And so clearly that's not what it is, right? It's about, like, internalizing that lesson, understanding where the pedagogy is integrated, but it's also using data to say, I am using this methodology with integrity, and it is not increasing the outcomes of these particular students. And so what are some potential barriers that are inherent in this program because the evidence base isn't a guarantee. So I want to tell you a story because I love stories and analogies and I promise this will make sense. So, one of my very best girlfriends, one of my college roommates, she was selling a very high end lotion. The company shall not be named. And she was like, you have to get this lotion [00:24:00] because In clinical trials or in clinical research literally, Katie, 90 percent of people saw like brighter skin and like less dark spots. I was like, okay, a number one, what are you saying? Like, I didn't even ask about this lotion. You're telling me I need this lotion. Right. But she's like, honestly, it's amazing. Right. And so I go, okay, good. That's awesome. Right. Like where I'm an Asian lady. And so as I age, I like to get a little brightness. So she gives me this. It is an, it is evidence based, right? Like, this lotion does work for most women. And she's like, you use it for eight weeks, and then I'll check in. And so I, I literally used it for eight weeks, I promise you. I used it every single day. And then, and eight weeks, she goes, well, did it work? And I was like, pfft, it didn't work. And she's like, well, did you use it correctly? And I'm like, well, that's a ridiculous question. Because how hard is it to put, like, serum on your face? And she's like, Katie. Did you use it correctly? And I was like, what do you even mean? What does it mean to use it [00:25:00] correctly? And it turns out like on the box and on the bottle It says very clearly like after using a cleanser in the morning and a toner and your own serum and this serum and lotion And I was like, oh my gosh. I didn't do all that. I didn't know to do all that. And I actually didn't have the resources to do all that. Because I don't have toner and I don't wash my face in the morning. I wash my face at night, right? And so all of these things where it's like I had something that was evidence based. But, I didn't truly understand what was required of me and what I find that a lot of the times teachers will be given what has the potential to be a really great instructional program. But they don't have the instruction up front, just like Ashley was saying, to really internalize. What is that actually going to mean for my instruction? What are all the resources that I need? How do I set myself up every single morning? And so what I loved is I was like, I don't want any more. And she goes, you can't [00:26:00] say it doesn't work because you never actually used it the way you're supposed to use it. And so I was working with a group of teachers recently and I loved them and they were like, So open minded and they're like, well, they keep saying we have to use this because of the evidence base. We have to use this because of the evidence base. And so I said, okay, so just really quickly, and it was a math program, I said, can you just talk to me about like, how much time do you have for math? On my honor, they're like, 50 minutes. I'm like, wait a second. Isn't that a 75 minute program? And they're like, yes, but well, then you cannot use it with integrity. Like, I honestly don't even know what to say to you right now. Your administrators are telling you that you have to use a 75 minute program with integrity in a 50 minute block. I feel, am I absolutely losing my mind, but. Like, that's the whole thing is we're setting teachers up for failure as if the PO for the program is magically going to turn 25 percent of proficiency into just to [00:27:00] 100%. And so Ashley, you can talk a lot more because I work with teachers in many different subject areas, but ultimately it comes down to, you don't just buy the program and magically get results. Just like me having the lotion on my counter didn't do anything differently. And so how do we really say, this is what it's going to mean to use this with integrity. And then to really say, do you have everything that you need to be able to do it? Not only in terms of time to actually use the instructional program, but like I have teachers who have brilliant instructional materials that are digital and they're not allowed to have kids on devices. The kids are required to use a workbook in an evidence based program with an adaptive digital component. And it's like, well the kids are too distracted on devices, so the administrators just want them in workbooks. And like, that's the type of thing we're hearing from teachers, and so it's like, we want to lift up the fact that teachers need to [00:28:00] have really great support, not only in the materials, but are you adopting materials that are able to be used with integrity and do people have the material to use them with integrity? And then ultimately, what is instruction going to look like? That's just first best, and then when the data indicates that even when those things are in place, if student outcomes are not increasing, what is the professional expertise needed to recognize? What are the lagging skills so that I can differentiate instruction? How do I address some of these misconceptions? And that requires, as Ashley said, a true understanding of what came before. Because it doesn't matter how good your fourth grade curriculum program is, if kids are coming up with second grade skills that you have to think about supplementing. So Ashley, would agree, push back on me? Like, what are you thinking about my analogy of the lotion? Ashley Marlow: am loving it and I want to expand on it because I want you, I want to illuminate your friend in this conversation because your friend was your trusted colleague, somebody who you could talk to about [00:29:00] skincare, right? Maybe has some understanding of your particular skin, maybe could look at your skin closely and help you identify what you need and then could guide you in how to use that tool to best meet the needs of your skin. So I would be remiss if we didn't bring up the collaboration piece because that is the other piece that is So critical that is lacking in professional development, the amount of teachers that I talked to that have zero conversations with their school based interventionists, zero targeted conversations with their school based special educators. That is a huge problem. You cannot universally design for every human in your classroom. If you cannot collaborate with the other experts in your building who may be able to guide you in what that student might need. in connection with your own expertise. That is the piece of the puzzle that I am so concerned about as I work in schools across the country that continues to [00:30:00] be to come up is it is impossible to design high quality instruction if you are isolated in a room alone. We have to continue to bring up the challenge of scheduling time. With our other content expertise, our special educators are interventionists so that we can actually design. You know, in your case, that was your skin care routine. But if we were going to actually design an instructional model that can meet all of the humans that are in our classrooms, we can't do that in isolation. Ross Romano: Yeah, what's imagining the bumper sticker in that? Katie Novak: Yeah. And what I would add too is like, here's the thing, and this is where some teachers are at, and I say this with love, Is I could have reached out at any time. I mean, this is my best friend and we text every day. I'm like, I already know how to do this. I'm good. And so like, what's really interesting is that like, I was already taught how to put lotion on my face, like back up sister, [00:31:00] as opposed to, and you know, again, I think that some people are like, I'm good. I was taught how to do this, but it's like, but this is a different regimen. And so, like, how do we really take the time to Unpack what that means. And so I think a lot of people will have questions and they'll say, okay, so how is it different then? How is it different? Right? Like, we all know what a traditional math lesson looks like, right? Is the teachers up at the board, the teachers doing the math, the students then copy and do it exactly like the teacher. It is silent in the room. Like, we're going way back, right? To it. Assembly line math of the 80s and early 90s. And we recognize, of course, that now calculators and machines are very, very good at that. And so we've had to evolve in a lot of ways and technology has evolved in a lot of ways. But Ashley, you're so great at. You know, talking about how does, let's say, let's just pick 5th grade, it's kind of right in the middle, it's kind of moving out of elementary into kind of [00:32:00] getting into some of those more algebraic concepts. But like, how does it look different if something is truly universally designed and the teacher has high quality instructional materials? Ashley Marlow: Okay, so let's picture us in fifth grade, right? So for many of us, that was a lot of stress because of all the things that happened in fifth grade. Right. And then on top of it, we, if we want to talk mathematics, we're talking fractions. Fractions are the most major content that we are addressing in fifth grade. So when I was in fifth grade, what that looked like was, that's why I brought up keep change flip earlier. Like I think I'm a little still scarred from it, right? Like you have to memorize this procedure. This is how, Oh, we just multiply right across who cares about the numerator and the denominator. Just. Multiply across, get your answer, and then see what you need to do with it. I had no clue. And it's funny that you specifically mentioned 5th grade fractions, because in most, in my case, and in most people's cases, that was how they learned [00:33:00] fractions if they were not a memorizer. Or they could not repeat the instructor's procedure, they were deemed not good at math. And unfortunately, what many of us experienced was middle school tracking. Many states had a sixth grade aptitude placement situation where you took that one test and it determined what's going to be the path basically for the rest of your life. Right? Like whether you can score whatever score you're required to get on this sixth grade test will determine whether you can get into pre algebra. Right? Which, as we talked about earlier, why that could be a huge problem, and who's always getting into pre algebra? All the people holding the identities that we would predict that they would hold getting into an advanced math class. Like, that's what we still see even today. Imagine that's what it used to look like, right? And so now, a fifth grade classroom that actually is using these best practices in teaching math in a universally designed learning environment would look like [00:34:00] We're going to talk about fractions within a relevant context. How many of us, I just had this happen, I'm like scooping, I was making pretzels for the Super Bowl yesterday, and I needed to make, I needed to scoop five cups of flour, but something happened and I only had a one fourth cup size scoop. Like, I don't know, my kids took them, like, that's what was left in the drawer was 1 4th. So I'm like, how many 1 4th size scoops, how many of those 1 4th scoops am I going to need to do in order to have 5 cups of flour, right? And so I'm thinking about how many iterations of 1 4th equals 5. Well, there's 4. Iterations of one fourth and one whole, right? So I have this context. I can wrap my head around when I would use this. So we're introducing a task like that. Something that would be relevant where your students can actually picture. Oh my gosh, that happens all the time in my house. We always lose our scoops like our measuring scoops, right? So we show those and we start by wrapping our [00:35:00] heads around a realistic situation, right? And then we start thinking about We start getting a chance to talk to each other. What is that going to mean? Am I going to have, am I going to need more than five of those scoops, right? So kids are talking to each other. We're explaining our thinking. We're wondering about things. This is all part of that main lesson, right? This is not like the warmup or like the, oh, this is just the fun part before we get to the real math. This is the real math because real math. Is math that can be, can help you to figure out what to do in your real life, right? In the real world. That is the real math class. And so we're talking about a relevant context. Then we're going to give them another situation. What would have happened if I only had a half size scoop, right? So that's a little bit of a friendlier fraction than one fourth. Now, I'm gonna send them off to go work together. But they're going to have access to the tools that they might need. We're gonna have scoops available. We're gonna actually, ideally, [00:36:00] have something they can actually practice scooping and compare the difference between the one, one whole cup size and the one half size. We're gonna have Fraction tiles available where students can build and iterate those fraction pieces until they get to five holes. We're going to have number lines that they can draw on graph paper. We're going to let them work together to come up with a solution, and then we're going to bring all their strategies back together. And we're going to put them up and this includes when we talk about ways of explaining your thinking, we're going to have kids being able to record their voices. They're going to be able to draw on the iPad, like, The goal here is to build our fractional reasoning. Not to be able to write in our math journal. Not, honestly, even to be able to write our thinking. Right? Our goal is to think about fractions. That is my goal. I'm going to stick with that firm goal, and I'm going to provide any of those ways that kids want to access it to match that goal. Because where we often got stuck [00:37:00] along the way when we used to teach math was I'm showing you a procedure and now you need to repeat it on pencil and paper. So now I eliminated all of the thinking. I didn't help you to wrap your head around how this is applicable to real life. And I provided no choice in the way that you showed me what you understood. In fact, I made the goal about writing your math thinking and repeating a procedure. Rather than having the goal to of actually understanding the mathematics. So a universally designed math class today includes relevant context. Students can choose the way to explore and engage in that context, and we are not going to sit in a one strategy approach anymore. We are going to allow students to share how they approach this situation. And then we are the facilitator, the guide of the conversation, the sense making bridges happen. through the conversation that we are facilitating by saying, Oh, [00:38:00] Katie got out the scoops and started scooping scoops of flour and counted how many until they matched the one whole, right? And then Ross built out half size pieces with fraction strips. What's the same about their model? What's the same about how they approach that? What's different about that? That is the instruction, not here's how. You solve this multiplication situation with a fraction. Go ahead and do 20 pages in your journal, and I'll run around putting out fires because everybody's either upset, not working, fighting with each other, or hiding under their desks. (ad here) Ross Romano: Yeah. Yeah. And I mean, perhaps you've already illustrated a version of this, or at least 1, 1 way of doing this, but to the point of cultivating, like, positive math mindsets, right? Like, an important. Piece of this and math, perhaps in particular, one of the subject where people will say they're not a math person or they're not good at math or, and [00:39:00] so again, within the dynamics of teaching and learning it requires that we have learners feel that they can learn and will learn and that they are capable of Being mathematical, being mathematicians I, I think your illustration that you just shared actually would be one way of that you could do an activity like that. And then afterwards say, look, we just did math. Right? But what are the other important. You know, ways to, to approach this, knowing that over, I don't know how long this has been going on, perhaps since math was invented. You know, that it still persists, right? That this is with every generation that goes through school and through life. This is still a prevalent mentality, which is a high percentage of people feeling like math is just not for them. Ashley Marlow: Yeah, in my experience When we [00:40:00] provide the opportunity for kids to experience mathematics the way I just described, we are sending the message that you were fully capable from the beginning. So, illuminating that story that I shared, my experience was I couldn't memorize the procedures, so I told myself I couldn't do it, so then I had to do this test that literally put me in a math class. It was called like remedial math or something, like the title even let me know that I was behind, right? So, All of those little messages are sent, and I'm not just talking about going into middle school. We see this happen as early as first and second graders. Saying, so and so is good at math in my class, so and so is not good at math in my class, I'm in the good math group. Like, this happens so early. So it's about shifting our mindsets and having real communication with students to disrupt, like, cultural. Cultural norm is, some people are good at math and some people are not. Like, the amount of memes that come up about, Math and [00:41:00] whether you're good at it or not, especially in movies, it's always the joke about like the like smart brainy one is good at math and like the cool cheerleader is not good at math. And like, that's a funny thing. You know, why can't the cool cheerleader be good at math? What if that was the trajectory all along? Kindergarten, preschool teachers are up against that. From the very beginning, so they have to deliberately provide those types of opportunities where kids are solving real math problems. We have to actually have conversations about this, that everybody is a mathematician because our job is to disrupt those messages that are outside of our classroom that we can't control. But in our classroom, we can control what that environment feels like, what it feels like to be a person who's doing math every single day. And I just. I think that if those are the types of experiences that students are having and having real conversations about who are [00:42:00] mathematicians, what is their job like, what are adults doing when they're doing math? I literally, I can't remember if we shared it in the book now, but I think I did. This, one of my friends is, he does sheetrock, and he was like, oh I hate math, I don't do math. And I'm like, Hello, you measure to like the nearest 16th, and your everything has to be level and 90 degree angles, like, you are using math all the time. Where along the way did we decide that that type of math, which is real world math, right, is not actually math? That math is really just what you do in a journal page. We've got to disrupt that. We've got to provide positive opportunities for students to actually do math. And we've got to provide opportunities for teachers to feel the same way. So going back to your PD question, teachers have to have positive experiences in math too, if they're going to be able to create that environment in their classroom. Because just last week, I was working with a teacher who said, I [00:43:00] want to tell you this, but I'm not really a math person. And like we're in a math PD session and I'm like, no, you absolutely are like, I am going to stick a sticker on your forehead. Like you are a math person because she hadn't yet had an experience where she felt like she was good enough in math. You can imagine the hurdle it would be for her to then create an environment where her students could feel differently. So. You know, I think that's like my life's work is starting to dismantle that for teachers and other adults and parents who are trying to do their best and help their kids. Because I just know that the world that I want my children to grow up in will need the diverse voices solving the problems that exist through mathematics. That are going to solve the climate crisis find a diagnose find a cure for cancer that will fall under people having creative brains, understanding mathematics. And so that's the long [00:44:00] winded response to a question about mindset and why we talked about it so much and why I think it's got to be part of our practice because I think it's the only way we're going to really change the culture. Ross Romano: Yeah, well, from a strengths based view the silver lining is if you are in remedial math and you know that that means you're not good at math, at least it means you're pretty good at English, Ashley Marlow: Yeah, right. Ross Romano: So that's not bad. Katie, going back to skin care Ashley Marlow: that's a great Katie Novak: Hold on, it took me like 10 seconds to get the joke. It was the definition of remedial. I was like, put it together, Katie. And I just, yeah, Katie just did the math and I got it. Ashley Marlow: Ross, I was in the Newberry and Caldecott Honors Book Club, so I had no issues with Katie Novak: Oh Ashley Marlow: confidence in Ross Romano: okay, well, keep it in mind for other kids. Katie Novak: Back to skincare. Ross Romano: but truly like jumping off of that there also is the piece of building the confidence of capable math [00:45:00] teachers and not just like capable of math teachers, but to the point of, you know. In the, maybe not to over leverage your anecdote, but if this was either like the fifth different skincare routine, your friend had recommended to you and every one of the ones before, like didn't really work or she was the fifth different person to, to tell you to do this and you're like, okay, like, all right, I'll give it a shot, but, I'm not listening too closely. Right. And it would be incumbent on that person to say, look, I understand that I'm not the first person to come to you with this or that in the past. You've been given these unreasonable routines and regimens that you couldn't possibly succeed with. But this 1 really can't be done. And this is the evidence for it. And here's how we're going to create the conditions for that to happen. Right? That's. Yes. Realistically, what's happening a [00:46:00] lot of the time, right, is okay, we have now landed on a good curriculum or instructional tool or whatever the case may be, and we feel really confident it's going to work. We've invested the money in it, but we need to create the conditions for teachers to succeed with that. And part of that is understanding that there probably have been a lot of other things that have been brought to them before and It wasn't really that realistic or it didn't succeed for whatever reason. And we need to be able to address all of that in in the process of making the case for why we need to really get on board with the new thing. Katie Novak: I love the extended metaphor and I'm totally on board. And I would also add to this is that I would be much more likely to do it if I had friends who tried it before me and were successful. So it's not just [00:47:00] The evidence says this, but look, I tried it. This is what I did. Here were my results type of thing. And what I am so passionate about is like a really good pilot before you go in and investing things. And I could talk about pilots forever. Let me talk about some of the things that we don't want to do with pilots. Hey, there's three programs. Choose one of them and pilot it. Okay, you're only going to choose one and that means two thirds of the people who truly invested their time into a program are now not going to have opportunities to persist with that. And so what we know is it's really more about like, before you even see start looking at programs, really coming down to what is it that we want for a program? What do we believe about kids? Who are the kids that we're serving? And ultimately, what are our non negotiables? It has to be something that is going to be really flexible. It has to be something that has both digital and hard copy. It has to come with a manipulatives. It has to come with professional [00:48:00] development. The teacher's materials have to be really easy to use, right? So actually creating a rubric before you start looking at things. And then having a group of a distributed leadership team of people in special education, in multilingual education, general education teachers to really think about, like, if we're looking at our math scope and sequence, like, if we were to try, you know. One lesson on this and another lesson on this and another lesson on this and to really observe what's happening with instruction to talk to kids about how they feel about the experience to look at their outcomes, right? And to really make a decision based on evidence to say when we use this, it was a really successful. We did see a difference. These are some barriers, but this is how our colleagues in special education helped us to eliminate those barriers. And I think a lot of the times, it's just, Hey, the state says we have to pick one of these five. Do you, who wants to try out one of them? We'll take a vote, but a big, but a [00:49:00] boom without really saying like, we need a program where the people who designed the program can tell us how to make this work in a 50 minute period, because it's all we have. And so I think the intentionality of the group of people thinking about, like, what are we going to use for materials, and is there some demonstration of impact that the methodology will be effective? And it's on a small scale, because obviously, like, that's what a pilot is. But you know, I recently was working with a district that because there was so much Conflict about which of the two programs that they piloted, because everybody piloted A or B. No one used both. And so then, the ones that you really get to know, with the vendors you really had a relationship with, you're like, I'll take that one, I'll take that one. That they literally then gave schools a choice. In a district with high mobility, where kids are going back and forth. Totally different program, totally different scope. And so I think that sometimes we really have to say, Before I just go out and buy it [00:50:00] because you told me to buy it, what are the other lotions that are available? And have you actually used any of them? And what else did you need to use? Right. And so like, again, it was just a funny little like, yeah, sure. I'll try it. Oh, there's a new lotion in that company. Cool. But I think sometimes we do. Curriculum adoption in the same way. Oh, sounds good. Yeah, hook me up. Another superintendent said it worked. And it's like, but what is it going to be able to look like here? And can we create that demonstration of impact? And then everything that you said with the culture of the truth is we only piloted this for a short amount of time. We are going to have to invest quite a bit in professional development. We are going to have to realize that it's not going to be a perfect system. We're not going to abandon it the moment that we feel like oh my gosh, it's not working. But I think we could play with this analogy forever. That before we even get the materials, have we really thought about, do we need those materials? And have we [00:51:00] tried them before we've committed to this wild 7 million purchase? Ross Romano: And actually, of course, 75 minutes is 50 percent more than 50 minutes, but 50 minutes is 33 percent less than 75 minutes. So, now you know I'm a math Katie Novak: Oh my gosh, Ross Romano: least as far as that basic math is concerned. But truthfully, I mean, to the point of algebra as the gatekeeper, I never I was always in high level math, but I never, I was, that was never the thing I was most interested in, so I never really considered myself that much of a math person, but then when I got out into the real world, and I was always around other people who really didn't fancy themselves math people, and realized that I always was at least very strong in algebra, right, and it made me, by comparison effective, and I I've never been in professions in which It required being super high level math, but there's a foundational, there's foundational [00:52:00] knowledge there that is. Is a strength. I, this next thing, I mean, there's like really no effective segue here, but it's just something I want to ask about. And so it's kind of, granular, but it's just something that has been trending, I think, as far as. What states are looking at important principles and concepts to be prioritizing their math instruction. So that's part of it. It is numeracy again, like, to, to listeners who are in a variety of roles and who maybe haven't been immersed in the components of. Math, teaching and learning, perhaps this, they haven't gotten as granular as to think about this on its own. But before we, we concluded here, I did really want to touch on the importance of, and my keys to delivering instruction around numeracy in particular. Ashley Marlow: Yeah, so are you talking about early numeracy, like pre [00:53:00] K? Or are you talking about the strand of the understanding of number pre K or both? Ross Romano: I don't know. Ashley Marlow: No, let's talk about it. I think that, well that's my, I guess. We can talk about them together because there's the challenge of early numeracy when we talk about pre K students for a long time we talked about reading a thousand books or whatever that number was, Dolly Parton's books before kindergarten thing. And now we really are reflecting on what are those early experiences three, four, and five year olds need in mathematics. And I love to say to pre K teachers, when you are working with students within five, and it's like one hand, right? You get to carry it around all the time. we're talking about within five, we are building the most critical building blocks that they will then need when they get to algebra in eighth grade. And those building blocks include, when I see five, do I also know that I see three and two? And that I [00:54:00] can also show that as three and two, right? And I know people can't see me doing the finger thing right now. It's my magic math hands that I like to do with young kids. But it's this idea of five is composed of multiple different ways of building it. And when we get to algebra, right, and you're looking at 3x plus 2y equals 24. You are thinking about what are the building blocks. We when we learned algebra, it was just how do I just work through a procedure to solve? But really, the thing that we're asking kids to figure out is what's the missing part? What could go there to make that statement true? In the same way that in pre K, we are thinking If I have three, and I have a an empty box, how many could be in that box if I wanted to have five? Or the game I play with kids is Bears in a Cave. So you have like little teddy [00:55:00] bears, and you can play within five or within ten, and you get a big cup, and you're like, Look at all my bears! I've got all these bears! What? Close your eyes! I'm gonna cover some of them up with a cup! How many are under my cup? Right? If you knew there were five and you see two there, how many are left under my cup? Right? That earliest entry point into thinking about the building blocks of numbers is the most critical mathematics they then need when they get into algebra. And so that's for my company, All Learners Network, that's a lot of the work that we do is we help people understand how does that concept then build into working within numbers within 20, right? And then we get into if I have 13, how many more to get to 20? How many, how Far am I from 10? What's the distance between 6 and 18? That's the earliest entry point into subtraction. Right? And so then as we move into second, first, and second grade, we're just doing that with larger numbers. What's the distance [00:56:00] between 138 and 429? Well, I can use I'm going to go back to my kindergarten make ten strategy. I can look for my tens and I can build to my hundreds and so on. And that carries into middle, into the multiplicative reasoning stage. When we need to be able to do that same thing when we're multiplying factors. When I'm multiplying three times seven, I spent forever, I would sit at the kitchen table practicing my sevens facts. I still get like anxiety thinking about my sevens, but if I understood that seven is really just a five and a two, that is a much better way for me to approach multiplying by my sevens. So we could go on and on talking about these key math ideas, but The reason why numeracy is a big part of the conversation now is because we know that students understanding building and breaking apart numbers, relative magnitude, understanding cardinality, being able to [00:57:00] match one to one correspondence match. You know, I have three. I have my five fingers on my hand. This is one. Right. But simultaneously, this is also one finger, but when I put them together, that is two fingers, that's like, that's a huge idea, but it is the most foundational mathematics. And how do we make sure that our youngest learners have access to that? And again, I'm gonna really ring that bell around. Do the teachers understand in eighth grade how that math in pre K provides access to that? To their eighth grades, eighth graders entering into algebra. Why was that so important? We need our teachers to see the connection in those building blocks across grade levels. If we want to actually design instruction that met meet students where they are at math with their math understanding. Ross Romano: Yeah. And I know this isn't this is certainly not the point of what you were just describing, which is you gave us the good knowledge, [00:58:00] but going back to what you referenced kind of early on in that answer. And it does illustrate part of the divide between, like, how much time and emphasis is sometimes spent on early math versus reading. I mentioned Dolly Parton, and I think it's 925 books that she recommends that kids read. I'm pretty sure 925, yeah. Ashley Marlow: And what if you were playing math games to write? Like, what if we were doing those games like the bear in the cave game or looking for numbers when we're driving to school in the morning? That really sets up our students for a different experience. But what we're up against is. Parents, caregivers, beliefs in their own abilities in math, right? We talked about that earlier. Those cultural ingrained beliefs in our own abilities impact how we show up as parents and teachers and adults in the world supporting our students. Ross Romano: you know, see listeners out there when I deliver it really dryly, if you're out there. So that joke was just for me. [00:59:00] So, this has been a really wonderful conversation. And I think, I mean, it's just foundational, honestly, to to learning and to students being prepared for successful futures. And in kind of closing this, I, Wanted to give you each an opportunity to share anything else you want to what's one thing that I didn't ask about that is important to be considered as part of this conversation or that you want to reinforce. I guess we'll go Ashley and then Katie. Ashley Marlow: For me, it's the message that I am constantly sending that I hope folks can listen to this and walk away with that. I know Katie talks about a lot in her work as well. We're not asking folks to do more, to add more and more and more to their plate. What we're asking folks to consider is the, is what you're currently doing. Actually working for the human beings in front of you. Because I don't believe anybody got [01:00:00] into this to be ineffective. Right? So is it actually working? And if it's not, could some of these recommendations? Try one thing. Maybe try more open ended questions. Try having your kids talk to each other more. Try playing more math games. You know, is there one change in your practice that will end up being more effective, that does demonstrate That you're able to meet the needs of more of your learners rather than this idea that we're packing more and more on because teachers are exhausted across the country. That is universal. And for both Katie and I know we both feel really strongly that we need to give teachers the tools that they need. And teachers need to then be able to decipher what's going to actually match what their students needs, and how are we part of that solution. So I think that's my last thing I wanted to say. What about you, Katie? Katie Novak: what I would add is [01:01:00] just we cannot ignore how efficient technology is becoming and this over reliance on like computation and memorizing formulas. I do a lot of very high level statistics and analyses of different districts I'm working with. And as long as I know where to go to find the right formulas, I can get those calculated. And then I use a lot of reasoning to figure out. Is it accurate? Because certainly you always have to like fact check, just like we want students to make sure you're verifying your work. But like, I have to think about like, is that result reasonable? How do I know collaborating with other people. And it's not to say that we don't like, obviously I need to know what formula to search for. I need to know what I want to do before to find it. But I think that so much of the. The value of memorization is becoming really antiquated. And I think that it's like, well, we [01:02:00] want them to memorize their math facts. And certainly math fluency and numeracy is important, but like, if a kid is confident that they can solve problems and resourceful and understands that there is a way to scaffold some of those skills, Then we can do really, really high level mathematics because we have those tools available. And so it's like, how do we, at an age that is appropriate, start to teach students how to ethically and responsibly use some of these tools that are allowing them to be more efficient? And calculators in no way, shape, or form minimized the cognitive load of really high level mathematics, and I think that's going to continue to be true. But how do we teach students that like they can solve problems and reason effectively and communicate and try to figure out things and this idea of having to do the same exact task over and over and over again, quantity over quality, which has been a lot [01:03:00] of what historic math has focused on. Is just going to be less important as our calculators and robots just become more helpful and more advanced. So the human element of being able to use words to describe, this is what I want to do. This is the problem I'm going to solve. What is the formula I'm going to need? Can you run that formula for me? Right? is really about the math practices, which is what it's going to look like at the highest level of engineering. And we're afraid to give students sometimes that opportunity to start building that foundation. And so I think that certainly people are worried about, oh my gosh, is this going to mean that kids aren't going to think for themselves? No, kids are going to be able to think at much higher levels when they can. offload some of the working memory that used to have to go to memorizing what the teacher did so you could echo it back. Ashley Marlow: Save a lot of copies, too. Not running off all those math fact foot papers. Ross Romano: yes, awesome. Well, listeners, [01:04:00] I hope you've enjoyed the conversation as much as I have. We will put the links below. You can find the book Universal Design for Learning and Mathematics Instruction K to 5. It's published by CAST. We'll put the link below where you can find the book or you can, of course, get it wherever you get your books. We'll also put some other links Edutopia article. That Ashley and Katie authored, Making Math Accessible for All Students, and that kind of was a predecessor to this book. So, great place to start if you want to read the article and then get more of an understanding of what you might get from the book. As well as everything we've talked about here. We'll also put other links below to the websites and various other resources. And you can read more about our guests down there too. So please do check all of that out if it's useful to you. And please do also subscribe to the authority if you're not already, or visit bpodcast. network to learn about all of our shows [01:05:00] Katie and Ashley. Thanks again for being here. Ashley Marlow: Thanks for having us. Katie Novak: Thanks so much. I'll be sending my bumper sticker your way. I Ross Romano: I look forward to Ashley Marlow: And maybe some cream. Katie Novak: great cream for you, Ross.