We're not going to get through today without doing some math though. So hi everyone. I want to talk to you today about school algebra. For me, algebra was beautiful. It's a satisfying mystery. It's like a puzzle where all the pieces come together in exciting ways. Is that how you experienced school algebra? Why are you laughing? Maybe for you it was a different kind of mystery. Maybe for you it was a chore or hurdle. We know as educators, we see lots of ways that algebra is a hurdle for students. And we worry algebra is a gatekeeper and some students struggle to get through that gate. And we worry. Research tells us that success in algebra correlates with positive outcomes for students. That it's a predictor of later math course taking and other academic measures of success. We see Algebra as necessary as an opportunity. Because of this, efforts at improving access have focused on oxygen. The efforts at improving algebra have focused on access and timing. Getting more kids to enroll in algebra and enroll earlier. And they're working. Students are taking algebra and taking it ever earlier in their academic trajectories. But still, we worry, success in algebra and equitable success in particular has remained elusive. And among those who do succeed, they often don't persist into higher-level mathematics or stem majors. In Delaware, like elsewhere, racial gaps and algebra achievement compound as students move along the academic pipeline. And here we see how this translates into calculus enrollment in one district that is not representative of the racial demographics of the district as a whole. So we worry. But also we wonder, maybe the problem is algebra itself. I don't know if you like me have a Google alerts set for when algebra is in the news. But algebra has been in the crosshairs. Some have argued for its reform. Others, its abolition. Saying that it is an unnecessary gatekeeper, that it is too hard or too abstract, or that it is simply not relevant for students that the focus on algebra is stopping schools from offering things that matter that are meaningful to students. But I wonder, and I worry to say that algebra is too hard or too irrelevant means it's too hard for some students. Focusing on the course or the content absolves us of thinking critically about instruction, about how changing, how about changing how we teach algebra to better serve all students. And particularly those who have been historically marginalized by and in mathematics classrooms. And really, regardless of how we organize math in schools, students will encounter the ideas of algebra. And when they do, we want them to learn that essential content deeply. We want for the instruction they experienced to be conceptually focused, to be culturally responsive, and to support them to become reflective and flexible mathematical thinkers. And so we wonder, when we look inside algebra classrooms, when we look carefully at instruction, what do we see? I wondered, and I worried. I looked in my research, I've watched over 200 algebra classrooms. Come join me. It's delightful. And I find that classes, while they differ in some key ways, look very similar to my own days as an algebra student, a teacher demonstrating how to do a particular procedure, students following along, taking notes, practicing those procedures, doing the odds for homework. It was like 30 years ago that I was an algebra by the way. Alright, so if we want students to experience algebra that is different, that is conceptually focused, that is culturally responsive. This requires shifting instruction, but this is really hard work. And again, I worry we're telling teachers to change their practice in big ways. But I worry that we're not considering where teachers are starting from or understanding the good work they're already engaging in as a beneficial starting point from which to build, which to bridge. What if instead, we thought about instructional improvement as bridging work, as working with teachers to understand and analyze the good work they are already doing in their classrooms. And through small adjustments, elevate and deepen that work as a path to richer learning opportunities for students. In my research, I think about what we know about how students learn algebra, about where it presents struggles for students, and the types of teaching that supports algebra learning. And I've worked to identify as Gary talked about and describe some key instructional practices that can serve us along that bridge. Creating pathways to those richer learning opportunities and hopefully to more equitable mathematics classroom spaces. So I want to illustrate what I mean by taking you back to algebra class for a minute. It's gonna be okay. Take a breath. Okay? So this is an example. This is what we call a system of equations. I'm gonna get real excited about this, I'm sorry, in advance, but this is a key algebra standard. And what you see up top are two equations and we need to solve for x and for y. And teachers will frequently walk students through this procedure. This one's called the elimination method, and it involves adding those two equations together. So two x plus x gets you three x, y plus negative y. They mysteriously go away. Six plus nine equals 15, and then we divide by three, we can get X equals five. Just trust me. Once you know what x is, then you go back to one of those two original equations to find y. Here you see we went back to the second 15 minus y equals nine. We do some subtraction and multiplication. We get y equals negative four. I don't have time to really teach this, but you get the idea, right, and we get an answer five comma negative four. So what happened here? Well, we did it, right, we solve the problem. We got the right answer. But what's missing? We missed out for important opportunities for rich and important mathematical connections to make sense of this procedure, to understand this puzzle, to understand why it works. What allows us to just add those equations together? Where did the y's go? We miss opportunities to support students to develop flexibility. What signal that the elimination method was a good approach to use here? Are there others we could've used. We might also have supported students to connect to mathematical representations. What does that five comma negative four. Where might we see it? What if instead we made a relatively small adjustment that starts here, but includes more attention to these ideas. Instead of walking through the procedure, we might, I know it's okay. We might instead rely on an instructional routine here we call this side-by-side worked examples. So what you see are three different students solutions to the same problem. And the middle one, which is attributed to Savvy, is the one we just talked about. But Nico and Elliot have two other ways of thinking about this. And we show them side-by-side. They're all correct, they're all mathematically valid. But each student has approached the problem in a different way. And instead of walking through the procedure step-by-step, we could focus our instruction on analysis, comparison. We might support teachers to develop questions that would guide that analysis. What allows each student to do what they did? What mathematical properties as each student drawing on when solving this problem? Which strategy is more efficient and why will it always be more efficient? Why do we get the same answer no matter how we solve it? When we do this, we changed the learning opportunities we offer to students. In answering these questions, we might highlight e.g. that Nico is actually doing something really clever. I know it looks long, but he's manipulating that second equation to get x by itself. And then drawing on the concept of equivalence to do something we call substitution. Eliot's doing something related but different. He's also manipulating, but in this case both equations and also drawing on equivalents but in a different way to set those two equations equal to each other. And then he goes on to solve, right? So when we shift this focus, we're creating space for this kind of analysis, for discussion, for seeing and noticing structure for flexibility. All the math coaches in the room are nodding right now for making important connections for reasoning. In doing this, we also create spaces importantly for more equitable mathematics classrooms, e.g. this routine offers the opportunity to position multiple students as mathematically competent. A practice that Dr. Johnny Wilson at UVA found as particularly important in classrooms were African-American students excelled. A teacher could highlight Nico obvious Eliot's contributions not only for their correctness, but for their use of key mathematical practices, making those contributions explicit to the class, positioning them as capable doers and solvers of mathematics. They might amplify Negroes use of mathematical structure and point out that that is something that mathematicians engage in. They might note zombies efficiency and highlight how efficient problem-solving is a really important mathematical practice. They might note Eliot's perseverance and cleverness here and manipulating two equations to get to be able to use a particular solution method. Now you may be wondering, how do we do this? Small change doesn't mean easy. Engaging in this incremental improvement work takes collaboration. Teachers need support to build these bridges. One way our research team does this is by supporting math coaches to work with teachers on cycles of planning, enactment and reflection, focusing on small shifts around specific instructional practices using routines like the one I just shared. We're also working to develop tools that coaches and teachers can use to engage in this work. This kind of incremental change also requires trust. Trusting that teachers have the capacity to do this work, trusting in their growth and development and investing in that. And it requires partnership between the University of Delaware researchers, state and district leaders, school leaders, partnerships like the ones we're sitting today, like the ones that the School Success Center can foster. These can help us bridge from worrying and wondering to meaningful changes that support student learning opportunities and ultimately their success. Thank you.
CEHD_Success Center Launch_Erica Litke.mov
From Robert Diiorio October 30, 2022
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