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Great books, great people, great ideas. Learning about these things is critical to being a well educated human being and we can help with the Hillsdale Dialogues. Each week, Hillsdale College President Larry Arne joins radio veteran Hugh Hewitt to discuss topics of enduring relevance. And from time to time they also talk about current events, but always with an eye toward more fundamental truths. And they want you to tune in to a conversation like no other. The Hillsdale Dialogues are posted every Monday on the Hillsdale College podcast network at par podcast hillsdale.edu that's podcast hillsdale edu or listen via Apple podcasts, Spotify, YouTube or wherever you find your audio. Welcome to The Hillsdale College K12 classical education podcast, bringing you insight into classical education and its unique emphasis on human virtue and moral character, responsible citizenship, content, rich curricula and teacher led classrooms. Now your host Scott Bertram.

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We continue a series of episodes from presentations delivered at Hoagland center for Teacher Excellence Seminars. The Hoagland center for Teacher Excellence, an outreach of The Hillsdale College K12 education office, offers educators the opportunity to deepen their content knowledge and refine their skills in the classroom. These one day conferences are hosted during the academic year in cities across the nation and feature presentations by Hillsdale College faculty, K12 office staff and leaders in the Hillsdale Network of member schools. There is no cost to attend and attendees may earn professional development credits. Currently, the Hoagland center is hosting a series exploring the art of teaching a variety of subjects. To learn more about upcoming events, Visit our website k12 hillsdale.edu.

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So when we talk about math at Atlanta Classical Academy, what we're really trying to get our students to see is the order, the beauty and the wonder of mathematics. So for order, we think that the world is a cosmos. There's a way things are, there's a way things ought to be. It's not always the same either. Sometimes the way things are doesn't match up with the way things ought to be. And that kind of grinds us and that calls us to action. We think that the order of the way things are that shows up in patterns. And patterns are beautiful, right? We love to see patterns. I played it really, really safe today. I'm wearing all solids with just a little bit here because if I had shown up with my striped shirt and my paisley tie and maybe some checkered pants, none of you would respect me because the patterns don't match up. It's not beautiful and you'd have just written me off from the word go. We love harmony. We love symmetry. We love patterns. That draws us in, and we think it's beautiful. And not only do we just get to sit and enjoy it in the moment, we get to do that. But patterns call us into questions. It calls us into the future. Where is this going? Can I make predictions about what might come next in this pattern? And then it also causes us to look back. Where did this come from? Can we dig a little deeper? Was this just random chance, or is there something lying underneath? And is that where this pattern comes from? And so I want to frame everything that we do this. And I'm not just doing that here with you. I do this with my pre calc students on the first day of class. I say, well, why are you all here? And we're in Atlanta. And so we always have a few students that want to go to Georgia Tech, and that's great, and I love them and good luck, because Georgia Tech is hard. But then I look at all the rest of them and I say, what are you doing here? And some students might give me some idea of like, well, math is useful in life, right? It could get me a job. And I said, look, we're going to do trig identity proofs. Unless you want my job, you're not going to use trig identity proofs. I'm the only one who ever uses them, and I don't plan on leaving, so they don't get my job. So what are you doing here? And this is what I lay out and challenge for my students. And that sounds great, but how do we actually do that then in practice? And this is where going to start trying to take some steps down from off of our ivory tower and talk about how this can actually be done in a math classroom. So we talk a lot at ACA about Aristotle's five common topics. And this is our structure for how we build a habit of questions. It can be difficult to come up with all the questions that we want to ask. So these are five categories of types of questions. The first is definitional questions. So things like what's a variable? If we're going to be talking about variables, what is it? Can someone give me a definition? Okay. We're using letters, and it grounds students. They know what they're about to do. Or if we're going to graph linear equations, what does it mean to graph? What does linear mean? All right, now I've built up a little something for what I need to do. There are also comparison questions, something like, how are squares like cubes? And then you can also ask, well, how are they different? You can ask things like, how is a radical like a fraction? Well, I can add 1/3 and 2/3, but I can't add one third and one fourth. Same thing. I can add square root of three and square root, like another square root of three, but I can't add square root of two and a square root of three. It doesn't work. So I'm starting to make connections to things that I've done. Relationship questions is less comparing one type of math to another type of math, fractions to radicals, two dimensions to three dimensions. But it's looking at the way things fit within a single problem. Y is the range of the sine function negative 1 to 1. Let's draw that back to. If we're trying to graph a sine wave, let's hearken back to, well, where did it come from? When we've got the unit circle, can the unit circle go up and down as far as we want? No. There's a cap to how much it goes up. There's a limit to how far down it can go. And we're asking these relationship questions. Or if I divide this, If I'm using Dr. Gregg's and I see 2 to the X equals 11, we'll just divide the two. What does dividing actually do? Is that the steps that we need? Circumstantial questions. What's the end goal? This is great for those second graders that don't know. They see two, they see four, they said six, they move on to pause and say, wait, what's our goal? Can we write an answer statement? And the answer statement says something like, we have blank ribbons in total. Oh, so did we find them in total yet? How am I gonna do that? Wow, there's more work to do. There's more than just one step. And then we can also ask questions like, man, we feel stumped. Is this even possible? We just got that two equaled five in this linear equation with x's on both sides. Can we even do this question? And then those can move us to authority type questions. Whether that is asking, do we have a formula that can solve this? Or whether it's saying, did we just come up with a formula? Did we just make new math? Is this a rule? Or maybe is this an exception? Right. Two plus two is four. Two times two is four. Does that mean that addition is the same as multiplication? No. So what we found is there's something special about two. And so these five common topics are a way that you can bombard your students with all different types of questions. If Your entire lesson is definitional, that it's going to be one boring class. You're just sitting there listing out definitions. If it's all relationship, how do you even know what you're talking about? You needed the definitional questions. You need all of these avenues to be able to work your way towards the heart of a problem. So how do we do this? Well, we need to start it strong. You need to model while you're teaching. And you need to say things like, I ask myself what's on the same side with the X? And then I ask myself, how do I get rid of it? And notice that you're telling students that you're thinking in questions. Everything that you do is. I'm wondering what comes next. I'm wondering how this part matches with that part. You also need to cold call. And it's uncomfortable because no one likes to be cold called. As I'm talking about cold calling, some of you are getting a little nervous that I might start cold calling on you, right? And so your heart rate's going up a little bit. No one really likes it. As a teacher in the first week of school, I hate that I have to cold call on the students because I don't necessarily know them yet. I don't know who's going to freeze. I have no idea what they're going to say. The younger, the less I know what they're going to say. But you have to build that up. Everyone has to know that they're a participant in the class because if they can go the first two weeks without ever talking, then they can probably go the first two months without ever talking. And if they can go the first two months, then they're going to go the whole year without ever talking in class, without ever actually engaging and doing mathematics with you. And that's a tragedy. So you need to be able to do that cold calling. And the great news is it only gets easier, right? When there's one cold call per class, pressure is on because it's the one time it's do or die. It's the only time I get this quarter to prove that I know math and I'm good at it. I. Or I'm a fool. But if you get three chances every single week, then you know that the pressure is actually off. It's actually okay to get things wrong. And the beautiful part of it is when you ask these different types of questions, find that brainiac kid that's always given you all the definitions and knows what to do and push him a little bit and everyone should answer something wrong from time to time, because then that's going to give students the confidence to say, well, if everyone can answer something wrong, everyone also probably is going to answer something right every time. And that's pretty encouraging. As the year goes on, you need to shift that responsibility, though. So you need to give up time to think. It's a really common thing, especially as I see new teachers, to think that if I am not speaking, if I am not doing math, then I'm not teaching, then something has to be happening. But thought is difficult, and you have to set aside the time for students to see that. When you say, I'm giving you two minutes, no one is allowed to give me the answer yet. But you have two minutes. You must think through how to get an answer. If you don't get there, that's fine. You still have to come up with at least one step. If you get there and it only took you a minute, you have another minute. You better find something. What are other things that you could think about? And notice, too, that when I give you that pause, that's when you started thinking. That's when you realized it wasn't just rhetorical. It wasn't just, I have the answers. And then you just have to sit and listen to my answers. When I paused, you took the time to think, what other things could you think about? When you have to answer and you may have thought, check your work, you may have thought, do it another way. The other thing is that you need to get to know your students. You need to know, again, who is that kid that can nail the definitions? Who is the one who is very, very literal? And you go, okay, I said this word. Here's our definition guy. He's got it. You need to know who's got the crazy out of the box? Sol. Who's just going to come up with anything? They're fearless. So when you have that really weird question and you need someone to come up with something way out of left field, you know, she's my girl. That's it. That's who I'm going to ask that question to, right? And this is how you build that whole community of everyone seeing their place in this habit of questions. I have a nice big list here of everyday questions. I think you could use every single one of these questions in every single one of your math classes. A couple of my favorite ones are, why does this one look so hard? Because everyone can answer that, right? Even if they go, that is the don't ask me how to solve It. But I can tell you why it looks hard. But the thing is, that's a really key insight. If you know what makes it difficult, you know where to begin, the problem. And especially if you've ever taught word problems, you know, figuring out where to begin is 95% of the battle. It is very difficult to figure out where to begin. And so that student that maybe lacks some confidence in how to solve it, they're a worthwhile part of the class. They've contributed, they know it. I also really love the can we undo this? That's a great sort of class to class. I know that one of my classes is going really well kind of in the long term when I can show up to a class one day and say, anyone want to guess what we're going to do today? What should come next? And if you've structured things well and you go, okay, well, we graphed sine yesterday, so what are we going to do today? And some of you are, you can't even help it. You have to say it because you're excited because you know we're going to graph cosine today. And that is exciting. And you should be excited about graph and cosine. Yeah. And so these are those everyday questions that you get to use. So I want to give you a quick maybe demo of this. We're doing great on time. I want to give you a quick demo of this for something on graph theory. Here's my. I have my pointer. I don't get to use these in class. So I'm going to point out here, I want you on any piece of paper that you may have, try to draw these shapes, but don't pick up your pencil. And now some of you are asking each other, some of you are, am I allowed to retrace the lines? Is that allowed? Let's say this is what I do with students. They want to know, what's the rule? What am I allowed to do? What am I not allowed to do? That's how we play games. Games have to have rules. So they want to know what's the rule. Let's say, no, you're not allowed. You are not allowed to redraw over the same line. You're not allowed to pick up your pencil. But sure, you can go through one point. You can cross lines. Let's call that ok. How many of you were able to do it for one of them? How many of you were able to do it for two of them? How many of you were able to do it for all three of them? How many of you think that the easiest one is this one right here? How many of you think the easiest one is this one right here? How many of you think the easiest one is that one right there? The majority of the room finds the star the easiest. Why do we think the star is the easiest and good? And if this was a classroom, you'd all have to be raising your hands. But you're all thinking and you're all adults, so we're going to just kind of move ahead. But what you're noticing is that you can start anywhere on the star. Start anywhere you can do it. The middle one. No one said it was the easiest. Do you think it's impossible, or did I just not give you enough time? If you think it's impossible, did you prove it's impossible, or did you just try it once and give up and throw up your hands and say, I can't do it? Did you try every possibility? How about on this one? Could you start anywhere you wanted? See, that's interesting. You can start there and draw the shape. Can't start there, though. You can start there. Can't start there, though. Is it because it's top and bottom? No, it's not because it's top and bottom. If I turned the shape over, it would still be true that it's those points. It would be side and side. What's going on is that those are three and three, those have odds. Look at my star. What's going on in my star? It's all evens. What's going on with my X in a box? I have odds and evens. Why do we think that this one is harder than this one? And do you see how now I'm going to be really mean and I'm not going to tell you because they only gave me 45 minutes and we all want lunch and so I have to move on. But that's mean, because what I've done is I've made you want to know. You want an explanation. And not only that, you're formulating one right now for yourself. You're coming up with things, and then once you come up with something, you're testing it, is it really working? Okay, well, we can do it by trial and error. And so we can spend a whole day trying to figure out how this works. And then what I get to do with matrices is talk about how this is an adjacency matrix, and we can call all of those different points a little section of my matrix, and we can say yes or no, and then this gets us into computer science and it gets us into catching mafia bosses and it gets us into the Facebook algorithm and everything kind of comes back to here to graph theory and, and now don't you want to know more about it? Because I didn't just tell you those things at first because I asked you a few simple questions. And so we have our building a habit of questions and we should do that every day. When I go in and I observe my colleagues at the math department our goal is that everyone get asked a question, every student is asked a question. Every single class period I've gone in and we have 50 minute classes and we usually do a 10 minute warm up and a 10 minute homework. So really we've got about 30 minutes for the lesson and I've clocked our teachers asking 50, 55 meaningful questions in a class period. That's amazing. And if you've ever read a Socratic dialogue you know that it's a lot of immediate back and forth. Socrates asks something, the participant says something, Socrates asks, the participant responds and then maybe it flip flops and the participant asks and Socrates has to say something but then he flips it right back around because Socrates doesn't like to say things, he likes to ask questions and he flips it right back around. And so I really think that in a lot of ways math is our most Socratic classroom in a lot of our classical schools. But what a lot of people think of a Socratic discussion as is more what I would call a seminar. This is when we get all the desks around in a circle and we ask a question and we debate it. Can we do that in math? Is that possible? And I think it is. And so that's what I'm going to call the deep dives. But I want to be careful with those and talk about when is it appropriate and when is it inappropriate to do those because they're difficult and if you do them too much I think you're avoiding that crossover that Dr. Greg talked about of too much open endedness. Well that's that fallibilist philosophy. Well you all have the right answer, so congratulations by the way. How boring is that, right? Just oh everyone, good job. Write something down. And you're right, that's boring. I want to be challenged, right? So appropriate times to do these deeper dives, maybe a whole 50 minute or however long your class periods are. A full on discussion. I think one appropriate time is the book ends. So I mean by that it's the very beginning of the unit or maybe the very beginning of the year. My next sort of example is going to be how I end pre calc and how I begin calculus. And it's one of Zeno's paradoxes because it's a great way to sort of cap things off. I also, my pre calc students, they're right in the throes of it. At the beginning of second quarter, they begin writing a pre calc essay about the ordered beauty of math. And so we have to have a discussion day where we talk about what in the world does that mean? And we need to discuss it and argue with each other. At the beginning of teaching imaginary numbers, I'll just ask my students something like, what is a number? Can you show me three? Can any of you, can you show me three? Now some of you are doing this. This is three, but it's not really three, right? Because if this is three, then none of the things that you did is three. This is just a reflection of three. This is maybe a picture of three. This is maybe an image of three image imaginary numbers. It doesn't mean that these numbers are fake. It doesn't mean that they're make believe. That's not what we mean by imaginary. We mean there's an interesting picture for them. If you know anything about the complex plane, you know, it's a really awesome picture where instead of a number line, we get to take numbers and say numbers get to be up here. That's an exciting picture. That's an imaginary number that we built this image to try and reflect its reality. That's amazing. This is how you can start a unit. And then you still have to teach them how to add, subtract, multiply and divide imaginary numbers. They need to be able to turn square root of negative 9 into 3. I they need those skills. You don't get to live forever in the discussion. But what a great way to start it or what a great way to finish it and cap it off. It's also good for those step backs. In February, the class comes in and you just see the deadness in their eyes. Maybe you've looked recently in the mirror and you saw the deadness in your own eyes and you realize, what is this all for? And in the midst of your existential crisis of what are you doing with your life? You invite the students into that and you ask them what they're doing with their life. Can math give us any insights into that? You can recall back to that order and beauty and wonder and you can say, well, why is February so awful? Or if we had to chart out the school year. Let's make a graph of the school year and then we can talk about graphs and we can say okay, we're in calculus. What's the concavity of January to February? And it is deeply negative because we plummet. Right. But March comes and even though F prime is still negative we're still going down. F double prime is positive. There's light at the end of the tunnel. We're starting to come back up. Even if we're going down we know we're going to make it back up. And you get to have this talk and you have that step back now. Inappropriate times are when you haven't planned it. The five minute rabbit trail tangent discussion that you thought you could turn into just a really engaging class discussion. Wasn't it was those 16 year olds being master manipulators. Yeah. That got the most nods of anything I've said so far that maybe even be a worthy thing to talk about that they've brought up. But you need time to plan it. So you might have to say that's awesome. I really want to talk about that. We can't do it today because honestly I'm not really ready for it. But then you go and you look in your calendar and you say next week, Tuesday we'll have finished our next test. We can talk about this. That's going to be great. Right? So those unplanned times also tell the students the plan. We also don't want them to think that all of a sudden you just forgot to teach them something that day and they just have this great discussion and they walk away going we didn't have to do math today. Let them know that this is maybe the most important math that they're doing is when you have these discussion days. And then another inappropriate one is when you need to be the expert. Because you are. Math has developed over thousands of years. You get 13. They're not going to make it without you. You have to guide them. Another of my favorite Ponkare quotes and I was reminded of it in Dr. Gregg's talk is the teacher's job is to make the student's soul pass through all the knowledge of his forefathers. Suppressing none. So you have to. You have to get it all the way through. Or suppressing some but forgetting none. That's it. Suppressing some. Some things we can skip over. We can hit fast forward but we're not going to get rid of anything. So the students need your help. And when you see those times when you need to be the expert, you need to step up and be the expert. We're doing all right on time. How do we plan this? It's through essential questions. This is the language that we use down in Atlanta is begin class with that one question that you can build everything else off of. When I'm doing my lesson planning these days, I spend maybe a little over 50% of my time simply trying to come up with this one question that will lead the class. Because if I've done it right, everything else is going to fall into place. So let me give you a couple of examples. One from math, one actually not from math, from elementary school spelling. So we could ask something. If we want to add with common denominators. A question like, what are common denominators? Is not an essential question. A question like, why do we need common denominators? Is not the essential question. Because if I start class that way, nobody knows the answer. But if I tell my fifth, sixth, seventh graders, I have two dimes, three quarters, how much money do I have? And they all know the answer is 95. How did you get 95 from 2 and 3? Maybe the 5. I got it 5 because 2 plus 3 is 5 and then 3 squared is 9. 95. Nailed it. No, what did we do? How did we solve it? Well, we said that two dimes is actually 10 cents. We said that three quarters is actually 75 cents. And so when we added the things up, you also didn't tell me five coins, though. You could have told me five coins. Would that be a correct answer? You could tell me five coins. You could also tell me 95 cents. And then you get to tie in Latin and you get to ask the kids, where does cent come from? And then they say, Centa is 100. And then you say that what they did was they turned that into hundredths. And so their answer was 9,500. That's why we need common denominators. And then you get to tie in more Latin because denominator, nomen, name. We had to give them the same name. And all of a sudden you get this great. Oh, my goodness. It started with a very simple question that everyone could get. But we dove into, what is it that we really need with common denominators? We. We can only add when things are alike enough. That's what we're trying to get at. We can also do one like this. Let's say we're trying to learn the. Is this Rule 6? Any of you explicit phonics? This is Rule 6, I believe, of our silent final E in spelling. Don't quote me on that, though. I put these words up on the board and these words up on the board. What do you notice? And there's not a kid in the second grade that doesn't notice something about those. Right? They can't wait to tell you the things that they notice. If it was me, they would notice. That's my name, right? And you go, that's great, buddy. And then you move on because that's not really what you cared about. Though maybe you could have a discussion of that's why the S is capital and that's the only capital S. And you can sneak in some. Some More spelling stuff. But then we say, well, let's say them out loud. Rid, R, D, Hop, op, Sam. But then when we do the other ones, it's ride or I D, hope O. Same. A. Nice, because I'm from Michigan, I really nail that A. And what they see is, oh, the blue ones all have an E at the end. Why do you think I made them all blue? Because they all have an E at the end, but they also all make the same sound. All those things are together. The E at the end changes the sound. And again, I didn't just present it as this is the law. Follow it. I presented it as, do you see the law that's there in the background? It was there the whole time. Did you take the time to stop and think and ask the questions and wonder, where did it come from? So essential questions, if I had to boil it down to two things, they need to be simple. Avoid math jargon. If I'm introducing common denominators, I'm not going to use common denominators in my question. That defeats the whole purpose. They don't know what it is. Don't use complicated vocabulary. This should be the thing where, again, if class is a discussion, everyone needs to feel like from the get go, I can contribute to this discussion. And you do that by using simple language as a total aside too. This is why I love classic ancient philosophy. This is why I love the speeches of our earliest presidents versus our later presidents. Go read the Gettysburg Address. There's something like three words, more than three syllables, some ridiculous thing like that. And yet it's so powerful. You can say powerful things with simple words. Do that in your essential questions. The other one is they have to be ponderable. No one can wonder if you ask them, what is the sixth thing an E at the end of the word can do? I don't know. Or if they can wonder something. All they're going to wonder are these fantastical has nothing to do with reality things. It needs to be something that they can wonder about. A question like how does exponential growth differ from linear growth is boring and filled with complications. Words something like, I'll give you a dollar today and $2 tomorrow and $4 tomorrow and 8 and 16. Meanwhile, I'm going to give this person 10, 10, 10, 10, 10,. 10, 10. Which deal do you want? You can wonder, you can start to get at what's actually the difference between these things. And then you also need lots of follow up questions. You need flexible routes. You start off on this amazing path and then they've got to take it over. Because if you're just holding their hands all the way through, then again you're not really asking questions. They're all rhetorical. Then they're not really questions you need to have room for. What are all the crazy ways a student could answer this? But then you need all of those to be directed back towards a firm end and at the end of class, or if you're really mean, at the end of the next class because you leave them on a cliffhanger and you don't tell them why. Graph theory works that way. And the one shape's impossible and the one shape is possible sometimes and the other shape's always possible. You just leave them on the edge and you say you're going to have to find out later. But you always need a firm ending at some point, right? They need something to hold on to. They need to know that the wandering isn't just about the wandering. It's also about finding those little nuggets of real truth along the way. So here's one of my favorites. I've had students. 1 and she went to tech. So she was a nerd and it was great. But she literally leapt out of her desk at this. Because it was just so. We have Zeno's Paradox, we have our tortoise racing the famed Achilles. And if you're at a classical school, then all the kids get why Achilles is the flash. Fleet footed Achilles, you see, who remembers how much of the opening of the Iliad. And then you move on and you say they're going to have a race. And your essential question is, who wins? Two words, two syllables. Doesn't get much easier than that. Who wins? And you would all say Achilles should win, right? A person versus a turtle. That's not fair. Can we make it more fair? How could we make it Fair. Give him a head start. That's a sane answer, right? 16 year old boys say cut his Achilles tendon and you go, well maybe, but let's go with the sane one and give the turtle a head start. And so we say at time zero, this is where they are. Okay. The gun. Not a gun. This is ancient Greece. The arrow goes off, right? Flaming arrow into the sky. The race has begun. What does Achilles do? He runs. So he wants to try to catch up. At some point he'll have to make it there. At some point he'll make it there. At some point he'll make it where the turtle was. Let's call that time 1. We've got someone stationed there. And they say, great, Achilles made it to where the turtle had the head start. But what has the turtle done in the meantime? He's moved forward, right? He wants to win two. He's already one and oh, he's beat the hare. He's trying to go two and oh, so he moves forward. Now notice. Did he go as far forward? Why not? He's slower. He's slower. I'm trying to make this as realistic as I can. Right? He didn't go as far as. So at time one, who's in the lead? Tortoise. Who do you think is going to win? I'm here. Okay, I heard a lot of Achilles. If Achilles wants to win, what does he have to do? You think he's not already running as fast as he can? I mean, this is Achilles. Does Achilles do things half heartedly? No, he does not do anything half heartedly. He has already run as fast as he can. So if he's already running as fast as he can, the only thing he can do is keep running that fast. Eventually he makes it there to where the turtle was. We have someone kind of stationed there to say, okay, yep, he made it to where the turtle was at time one. But what has the turtle done? He's no fool. Move forward. Who's in the lead? Tortoise. Who's going to win though? Some more mixed answers. Okay, Achilles wants to run. He's got it all tied up. Except it's not tied up. The tortoise is in the lead. Who's going to win? And now either the class has gone silent. I've convinced the class that a turtle is faster than the Flash and we're at an impasse. We're at what that question was trying to get to, which is I want you to be stumped about something. I want you to have to question, will Achilles Ever pass the tortoise and how? These are the follow up questions then, right? My starting question was just who's faster? But everything going into that is. Well, okay, Sometimes students will argue about why Achilles is going faster. They use some fancy physics stuff. We don't need physics and math, but they use some fancy physics stuff, right? And they tell me that velocity is distance over time and they do all that fancy stuff and I go, great, you just made a great case for your argument. But what about mine? You didn't actually address my argument. So is it the same? To argue for you is against me. Well, there's an interesting discussion, right? How about what's a third plus a third plus a third? It's one. It's three. Three. That's one. What's 0.3 repeating? Plus 0.3 repeating plus 0.3 repeating? 0.9 repeating. So it's 0.9 repeating equal to one. What comes in between them? What's the. Oh, I got rid of this one. What's the closest number to one without getting to one? Sounds really easy, right? I did use a two syllable word there. Closest, but sounds really easy. Gets you stuck in all sorts of things like denseness and compactability. And did you want to start talking real analysis? Awesome. Dr. Gabler is going to teach later this afternoon. He would love to talk real analysis with you. So what we get to is what's true about adding infinity? If I add up infinitely many things, is it always infinity? It seems like we can actually get somewhere with infinity. Welcome to calculus. Right, where we play with infinity. What a great way to get introduced to it, right? What a way to feel like. Well, I got to figure this out because either Zeno is right and all of motion is just a combined illusion and the world isn't actually in motion, or else we can do some pretty weird but powerful things with math. So to end things, I'm two minutes over. To end things, I have just a few recommended resources places you can look. The first is you should absolutely get a copy for Mathematics of Human Flourishing. If you already have one, get one for your friend. It's a lovely read, It's a very accessible read and you should absolutely sit with it for a while. Another one that also Dr. Greg quoted from in his talk is Beauty for Truth's Sake by Caldecott. That one gives a bit of a history of an apology for the quadrivium as a whole. So math's place, but then also all of the mathematical arts, how do they all fit into a cohesive worldview. If you want some fun, engaging videos of someone who's fairly mainstream and also kind of follows this math is wonderful, and let's just dig into how great it is. Three Blue one Brown is run by a guy named Grant Sanderson. It's a YouTube video or a YouTube channel, and he has some amazing explanations for questions like imagine you have a sphere. You take four points on the sphere. You build a pyramid between your four points. What are the chances that your pyramid contains the center of the sphere? So simple and the explanation is so elegant and yet all of you are racking your brains to try and figure out, well, what could it be? And you're bringing in all sorts of different math. So he's a master at that. And then lastly too, I would love to answer questions.