 So the question is, why do teenagers study mathematics? We have some sense that they do it for application down the road, maybe because they're high school or the college they want to go to requires it. Maybe because they have a test coming up, but we also like to think that there are other reasons too. So in order to start thinking about why teenagers study mathematics, I wanna think about why teenagers study some of the other topics that they study in high school. And we're gonna use that to help guide us and lead us to an answer about math. So here, Hume is telling us that we study history because it allows us to think about universal conditions, things about human nature, particularly common threads through human history so we can better understand our place in history. And so we do this with high school students. We let them see empires, we let them see wars, they analyze the history of mankind. And through all these things, we try and get them to trace threads so that they can understand the present, big ideas. In English, we do this as well. C.S. Lewis is telling us that literature actually adds to our reality. And so hopefully, when we let students engage with difficult literature in high school, it can enhance their sense of reality. And we do that too. We let students read things that are incredibly complex, that deal with issues that might be beyond their years, that might seem really challenging for them to engage with. Science. So science allows us to figure out how the world works around us and it gives students tools for interpreting and understanding that world. So the goal of science for students is that they can use these laws to interpret the world around them. They can understand where we came from. They can understand what's happening to our world. And they can understand the laws that govern our physical world as well. Big ideas. So to math. We're here to George Cantern. He says the essence of mathematics lies in its freedom. I think what this is getting at is that math can be a language to describe other things, but it can also be a language in and of itself that possesses beauty and significance. When I look at these topics, all of them very important, things that students need to know, I'm not entirely sure that I see the freedom. And I'm not entirely sure that I see the big picture and the beauty. So we're gonna take two slides for everyone to reflect. Are these the things that you personally think about when you consider the freedom, beauty, and power of mathematics? So if your answer was perhaps no, let's think about some things that might be. So one of the first things that people learn in college is this thing about the space between zero and one being as big as the real line. That's crazy. I think that's an idea that high schoolers could really latch onto and it could be really exciting. You can do algebra with shapes. I remember in my first abstract algebra class when we learned about dihedral groups, I almost fell off my chair because I thought algebra was something you did with polynomials, not with shapes. It was incredible. You can represent algebra geometrically. And so the thing on your left side is from a math blogger video about the Pythagorean theorem. The thing on the right for my teachers, we've become very familiar with this. It's the Euclidean algorithm. It's continued fractions. We can also represent English as math. So the subject of Symbolic Logic aims to do just that. So we can take English sentences in the ways that we think about our speech and we can actually sort of mathetize that. I think that's pretty exciting. So what do those things have in common? I think on the left, we have something that can be right or wrong. And what I want to say is on the right, or those topics, are ways that students can have different types of competency and they can engage with the material in ways where it's not just right or wrong. PCMI is really important. I think that this is a community full of people who really want students to be doing math where they can show competency. And it's really important that we're all here and we're doing this work together. So I'm going to grad school next year. So my research interests include trying to find ways around math anxiety, trying to create more equity of opportunity, and trying to get rid of rote test-driven mathematics. My positive research interests are open-ended questions, connections between essay and proof writing, bringing deeper concepts from these subjects into the high school curriculum, and also pruning unnecessary and irrelevant topics along the way. What can you do? Work to curb math anxiety at all levels. Show students that math is fun. And finally, my most important point is to trust students. We trust students with history. We trust them with literature. We trust them with science. We need to trust them with math, too. Thank you. Thank you.