This episode is brought to you by Canopy. Canopy.us.classical. Hi, welcome to classical stuff. You should know a podcast about the classical world, ancient stuff, old things, education, books, art.
And but definitely not math. We don't do that stuff here. We stay away from that. Math is lame.
We don't touch that garbage. Hey, math is modern. Math is a modern invention. And nothing new with anything is less than 100 years old.
So I never saw a triangle before. That's right. What, 19? What year is it, 1921?
What year is it? You're the trine. My name is Graham Donaldson, and I'm here with my colleagues. Thomas, maybe.
Hi, Bermuda. And AJ Hanenberg. That's actually pretty good. Sorry.
This is Thomas again. I'm not AJ. You want to say hello? Sorry.
Hi. And today we are learning about math, I think. Is that fair, Thomas? No.
I'm always very unhelpful in my titles for these episodes. So no, the subject of my episode is anxiety about giving this episode. So hopefully you'll appreciate this. OK, so I feel like I made this joke with both of you in preparing for, in talking about what I was going to discuss today that I felt almost like I was threatening you all when I would say I'm going to do an episode on math, because I think most people don't find that topic interesting.
I know you two are very kind and supportive people, but I could also, I would not blame at least a listener who looked at the title and was like, I don't want to hear about this topic. So we'll get in a roundabout way to my topic, because that's the only way I ever enter into any of my topics. So gentlemen, let's start somewhere else before we get into maybe the meat of the episode today. I've been thinking about changing all of your classes.
Cool. So I know that you all currently teach 9th and 12th grade English. Yeah. Combined.
But I think I have a better way to do it. So normally you all teach. So 9th grade you read old books, 10th grade you read. That's super lame.
Yeah, super lame. Me books. Less old books and then 11th grade you read not old books. You're American books.
And then 12th grade you read new books. They're a mishmash in there. Oh, you read a mishmash in 12th grade. Well, then I don't like that.
Mitch match. Thank you English teacher. So here's my plan for a new English curriculum. Are you ready?
9th grade we're going to teach grammar. Now only the rules of grammar. We're not going to read long passages at any point. We're only going to teach the rules of grammar.
They're going to memorize the Latin phrases for those different forms of grammar. And we're just going to hammer that home every day for a year. That's fantastic. My job is way easier.
That's why. Oh, wow. That'll tie in later. It was good.
In 10th grade we're going to do an entire year on spelling. Now, again, we don't want any long passages here. We just want the most difficult words that no one uses. So like complicated words that aren't actually very helpful.
But that would prove that they've memorized the worksheet. I can't stand them. Yeah, good luck on that one. I have literally zero guess as to how to do that.
In 11th grade, so you know how they're different like genre of books and poetry? So we're going to have them memorize definitions of those different genres. But we don't want them to read those genre. But we want them to know the difference between fiction and nonfiction or a biography and nonfiction.
Am I going to show them examples of fiction and nonfiction? We need to be able to identify them. No, we wouldn't want that. But so they just need to learn those words themselves.
It's like a really good categorization system is what we care about. But not being able to read a work and know what it is. So would I give them the definition but slightly reworded so that they would have to say that's still the definition? That's exactly right.
And that's what the test would be actually. It's not being able to identify which one is actually correct definition. OK, then there's going to be a unit on fonts. Ooh, I should have done that.
But I didn't. I would love a font unit. How do we have an elective on fonts? Oh, that'd be great.
So you start with Sarah for sans-serif. And then how do you tell your helvetica from your work your way up to papyrus? Yeah, the final. Ariel is the finest of fonts.
No, I was saying how do you tell your helvetica from Ariel? It's a fair. That's a type of choice joke for all your designers. Thanks for the two of you.
Is that the end? They're almost identical. Are they actually? Yeah.
One is just trademarked by Apple. What do people love helvetica then? Because it's trademarked? I don't know.
OK. OK. Twelfth grade. So that would be an elective would be on fonts.
Right? Literally just identifying fonts. Not actually writing anything in those fonts or reading anything in those fonts, but just like literally. Or making their own fonts.
How dare you? How dare you? They're not ready for that. And the Twelfth grade is a really exciting one.
So as you all know, Mortimer Adler has a list of 102 great ideas that he kicks off the great books of the Western world with. So what students will do their twelfth grade year is they will get a dictionary and they will define each of those 102 words. And they're going to write that out by hand over the course of the entire year. So they're going to define what those great ideas are.
In a serif or a sans-serif script? It's a very important difference. And that's what they're great hinges on. OK.
So grammar, spelling, they'll learn how to categorize things. And then they will define very important great ideas. But not actually talk about them. That's miserable.
No, no, no. Well, Adjard is easier to teach. Because it's really easy to grade. It would be really nice.
That's true. And you know if they got the information. Because if they know one font first, the other is pretty obvious. If they can spell it or not, it's pretty obvious.
So I don't understand. It doesn't make the homework great. Why would this be bad? You're right.
It does not bad. Wait. No. You're not supposed to fold so easily.
No. I'm into it. No, stop. The first time you agree with me is this.
Wait. No. I'm starting email before I write it right now. Yeah, Thomas said it's a really great idea.
I'll be doing this for a year. This would be wonderful. Why is this a bad way to teach English? Because it's not actually English.
It's not actually English. You must not have heard me. It's grammar and spelling and categorizations. It's English.
What do you mean? They're definitions. Yeah, those are in English. They're by the English language.
Yeah. So it's all in English. I don't what? Why is this not English?
Because it's not really engaging with the practice. Yeah, the practice. Sorry, what? What's the point of English?
To feed our hearts and minds. Oh, wow. I hope all of your students who are listening. Obviously, all of them are listening just to be clear.
I hope they responded immediately with that answer too. That's from your catechism, correct? It is. That you read at the beginning.
Why do you study English literature to women? That's not the answer. I don't know what you are teaching them, but I watched. What was that?
Oh, Captain Mike Captain movie? I watched it. I was supposed to say it. Full point.
No one wins. I don't know what women are going to do with my English class. I don't even know why they're in there. You have to say that.
Yeah. Why do we read poetry? So women, this is unfortunate. He was a great teacher.
He got fired after a year. He got fired after a teacher. He got fired after a teacher. At some point, do you defend his book that they tore the page down?
Yes. I think JL for Richard's talking about like talking about. They give you the same method for evaluating poetry and perinze guide to literature, which I teach out of all the time. It's actually kind of useful.
This is a bad take. I don't like this at all. OK. So take it back to grammar spelling, categorization, and dictionary entries of defining gradadias.
Why is this a bad approach to English? You say it misses the point of English. But why? What is it missing?
They're not actually pushing off from the shore and set sail in a book. They're not actually going through what English creates. They're just learning the tools. You're ready for a little drop of logic here.
Oh, yeah. It's missing the essential attributes. What's that? So when you define something in logic, a definition is required to give the essential attributes of the thing.
So for example, a backpack, I could say all kinds of things about a backpack. It has zippers. It's a canvas. It can go in a trash can.
I can say all those things. But the essential attributes are that it's a bag and it goes on your back. It's a backpack. And so having them work on everything peripheral to the actual practice of English is missing the essential, which is communication, crafting of sentences, and then communicating great ideas.
So if they never craft sentences and communicate great ideas, they have missed the essential attributes. And so I have a feeling about where you're going with this. I'm not going to jump. I won't cut you off.
You drive that bus. That's not true. You're a big way to go with this. No.
Where am I going with this? No. Your point is that the math curriculum currently doesn't do the actual practice of math. It doesn't do the essential attributes of math, which is, if I'm thinking about math correctly, is to, when they were originally discovering it, they thought it was the basic language of the universe and allowed them to do things they couldn't otherwise do, like measure things that weren't incredibly high.
And create cathedrals that stood even though they didn't seem like they should be able to, and to create artistic works. Yeah. I thought about it because you talked about cathedrals in the Gates Paradise episode, right? That which one was it that was started before they could put the dome on top of the Duomo in Florence.
They lacked the mathematical ability to approach that problem. My kids universally loathe geometry. But when I show them a book I have by Albrecht Dürer about the actual practice of geometry, he's creating fonts. He's showing you how to make rose windows, how to make monuments and sculpture, and how to actually set it up so that a design on a column will look the same size at the bottom as it does on the top, even though you're viewing it from much further away.
It's a geometric change in the pattern so that it fits the viewer's eye. Like all of those things on the practice of geometry. But our kids never get to do it. And same, Graham, I'm sure you've referenced it on this episode before.
You have an interesting math problem that you like to present to your leadership class. I do. I present to them. It is affectionately known as the murder circle.
But what it is is it's the story of, what's his name, the Jewish historian Josephus. Josephus was part of the rebellion against Rome and he was in one of the, a siege fortress. I think it was, I don't wanna start with the M, I can't remember. And so it's, and then, but they were gonna die.
Like the Romans are gonna get them. But Jewish law prohibited them from committing suicide. They didn't wanna get taken by the Romans. So what they decided to do was go sit in a circle and you would kill the person to your left.
And then the last person would either surrender or commit suicide. Only one person was committing a grievous fault, not everybody. And Josephus apparently like did the math quickly in his head to figure out where he had to sit. So that he would be the last man standing and then surrender to the Romans and ended up becoming best friends with the Roman general who took over the fort and that Roman general general one day became emperor and Josephus was saved.
The great story. And so the, so there's an equation that you can get to of figuring out. So like the question as my students is, if n equals the number of chair, a number of people in the murder circle, what is the basically at the F of n, or the L of n, of where you need to sit so you don't get murdered? So you're the last person standing.
And we spend some time whenever, this is I do this in leadership class whenever we don't want to do the other curriculum. Can you say that? Are you allowed to record that? I mean, here we are.
And there's a way to get to it. But I never tell the students the answer. I just let them try to figure it out for the hour and then they usually leave. And then we'll just Google it and then get it solved for them.
And that was just kind of annoying. But some students will figure it out. It's complicated. I was wondering if do they get to an answer during class?
Yes. They figure out pretty quickly not to sit in an even number chair. So if one is the first person, one kills two, three kills four, et cetera. So don't sit in an even number.
And then there's like the number ends up being something like I can't remember the equation on the top of my head. But there's ways to do it. And would you even be able to think about whether that belongs in is that an algebra problem, a geometry problem, or a calculus or a like is there any way of even algebra? That was just one.
It's a very practical problem when the Romans are coming up before it comes up before it comes up. They're like bro getting the circle and you're like, well, one second. You're like a chalkboard. You're like, OK.
I'm like, oh, I'm a think it's out. So you had your bowl. You kill you and you kill you. What's he doing over there?
You're in for this right? Oh, yeah. It's a great plan. All right.
So I'm coming up this from an essay that I know you all have read, but it's been a very long time. So you obviously will remember all of it. A mathematician's lament. Is this a name that rings about?
No, does not. Yep. Do you remember the last time that there was a Magistorum held at this school? That was my first start.
It was my first time. But a mathematician's lament started out as an essay written by Mr. Paul Lockhart. It's from 2002.
So it's not classical. So please tell me about how we're not actually classical stuff. You should know. It was eventually expanded into a book.
And if you want to read more of his thoughts on education, you could read that. But I'm focused on the original essay. Essays from 2002. The book is from 2009.
And I was trying to look up information on Paul Lockhart. And there's not like a ton out there. I would love if he'd listen to our podcast that actually make my day. But he taught.
I'm not sure he currently teaches there. But at least until 2016 was at St. Ann's School in Brooklyn. Might still be there.
He published a book in 2019 on Elderep. So and I found nothing else about the guy other than that. So he's a math teacher. That's the important part.
And he wrote an essay about the problems of how math is taught. And that I think aligns with why I had this nervousness about talking about this topic on the podcast. Because I don't want my episodes are boring. But I'm not intentionally bringing a boring topic to the table.
And I think my apprehension comes from most people's experience of math, which only really happens in school. Most people don't go home and solve proofs for fun. Though if you're on Twitter, Mr. Nussim Taleb, I think every Sunday solves a proof.
That's his for fun activity. I just can't understand it. I just realize it's. What do you mean?
I mean his proof are things I don't know. Oh, yes. Who is this guy? Nussim Taleb?
He wrote a black swan is probably what he's most known for. Anti-fragile. Skin in the game. Famous for Magic.
Are these Scientology books? Yep. Everything on one of them. Where?
This is a Scientology podcast. Wait. Let's see. A very infamous trader who made a bunch of money in shorting the market right at the right time.
Lebanese. I believe Lebanese? Family fled Beirut during the war in the 80s. Kind of lost everything.
He was in New York. Made a money on one big trade, but had been consistently losing money for a long period of time, losing little bits of money and then hit it and was basically going long vol on some trades. That means long volatiles. Long volatiles.
He's betting on the volatiles. And then one big win, the thing that was eventually going to happen, what was going to happen? And his whole point was we look at the world and think the world is sort of the way it's been going as a way it's going to continue to go. And we discredit and disvalue volatility and rapid changes.
And the smart money bets on those rapid changes and has the iron stomach to withstand the small losses until you hit the big payday. And then wrote all these books and is notorious for getting on to fight on Twitter. Yes. These are all true things.
OK. So that is, again, the statement of why was I nervous in presenting this? I think most people have a bad background with math as a topic. I don't know if this describes you, though.
How do you all feel about the math feels like too big a thing to have an opinion about? I guess think back on your experience in high school, maybe college if you took math there. What is your background with math classwork? When I was in elementary school, I had to do math in French because I was in a bilingual program.
And I didn't know French very well. And so I could do the last math lesson, teach it, and I could do the lesson. And then when we got to the word problems, I didn't know what the word problem was asking me to do. So I was just looking at the word problem, and I see that there was an eight and a three, and then a bunch of French words that I didn't understand.
And I was like, well, the lesson was multiplication. I was going to multiply those numbers together. It worked. Like maybe 40% of the time.
And so people thought I was dumb because they were like, oh, he keeps getting these answers wrong. But I was like, I can't read the language. That was my elementary school. And then eventually I figured it out.
And I didn't do math in high school. But I didn't do French math in high school. I didn't do French math in high school. And I always enjoyed the logic puzzle problem of it.
My dad did his undergrad in chemistry or physics. And so he was very mathematical, and then went into theology. So he would always help me with math homework. And I would always remember him getting not frustrated, but he would say things like, oh, it looks like the book, or it looks like your teacher wants you to do it this way.
You could also do it this way. And to me, that had such an impression on my brain that a math wasn't math, math was compliance. And that made me frustrated. And I didn't want to have to.
I hated not that I hate compliance. I just sort of, I think every lots of kids look at math and say, oh, why do you make me do it in this sort of systematic approach? Why can't I do it this other way that makes sense to me? But you get more grades taken off and you get frustrated.
So that was sort of my experience with math. Fellas, the intellectual life is a worthy pursuit. But in the modern world, there are lots of distractions. I mean, the internet is literally a machine that is dedicated to stealing our attention.
That's hard to keep that intellectual life if you've got something that is constantly stealing your attention. There's been tons of apps that have come out that block websites or limit-restrachioned websites. But this new app called Canopy, which is a sponsor for today's episode, is awesome. Not only can it block entire websites that are big time sucks, like Facebook, Instagram, Twitter.
But if you're a parent, it can also block explicit content for your student. If you're a school, it's something that can not only just block entire web pages, but it can block partial web pages. So if there is a web page that has worthy content, but has questionable advertisers, Canopy can block that stuff out. Yeah, it's great.
And the vast majority of Canopy's use case scenarios is on mobile, because studies have shown that most teenagers now interact with questionable and explicit content on their phones. I remember when I used to teach at a school a bunch of years ago, we tried to do one of these first generation website blockers. And kids could get around it in 30 seconds. But with Canopy, the parental controls of it are top notch.
There's the parental app, and then the app that goes on the phone, students can't get around it. They can't access web pages through Google Maps, which is even a thing that you can do. They've thought of everything to make sure that any kind of front-facing or any kind of web browsing application is going to have Canopy integrated with it. And you guys that listen to our podcast, you can go to canopy.us backslashclassical.
And you can get 30 days for free and up to 20% off forever. So you can sign up. You get a 20% discount forever. That's great.
So canopy.us, that's Canopy with a C.us backslashclassical for 20 days free and 20% off forever. I loved math. Really? Oh, yeah.
Cool. This is you were a math lead. Yeah, I promise this doesn't sound like some humble breaking. They shipped me off to the high school in eighth grade, so I could take high school math classes.
And I ended, I finally finished with my math career when I took pre-cal as a sophomore. And I was like, I just don't think I want to do this anymore. I felt like math had become impractical. It wasn't for the real world.
And I didn't think I wanted a job where I was doing math all day. So it seemed like I'd reached the end of my necessary math. But prior to that, I had been going to competitions. I think I won regionals and got second state at some point.
But I always loved math. And so I remember this essay. And I don't know. I have certain feelings about it.
I'm sure what will you think. Yeah, well, we'll get there. They're in practical parts to do it, because it's more of a germinate. It's just a condemnation of how math is currently done without much of a bunch in the way of where to go forward.
We'll get there. So my example of changing the English curriculum comes from this opening, oh, sorry. I guess for background, I don't have any strong memories of math. I have good teachers and I have bad teachers.
I took math courses all through high school in college. I took extra probability courses, probability stats, math courses in college. I just enjoyed them. Almost got a math minor, but didn't and more out of like I had to graduate at some point other than, yeah, so nothing really.
I've taken math. I enjoy it. So I don't have maybe these negative associations, which again is what made me want to look into the history of the subject, which we'll get into at some other point in a different episode, not this one. This one is just about me not wanting to talk about this topic that I find interesting.
So go into the essay. We'll talk through, we'll give a summary of it. All of my episodes are book reports. The opening story in Lockhart's essay is about a musician waking up from a nightmare.
And the idea of this nightmare is that the way music is taught is that students don't get to create music at any point in their musical education. It's all about theory. It's all about reading sheet music. It's all about changing the key, transcribing music into different keys.
And that's how you learn about music. At no point do people actually get to do music and make mistakes. It's very boring. He wakes up in a terror and then a painter starts to have the same series of nightmares.
In my example about English, the thing I just kept thinking is that it's like AJ. When you're talking with ninth graders about the Elliot or the Odyssey, in some sense, aren't they unprepared for the Elliot and the Odyssey? Yep. Like there are things going on in that book that they won't understand.
And they won't even understand it by the time they graduate. Well, maybe. But I guess the, so it's easy to make fun of, like, obviously a year of grammar, well, grammar's a classical subject. No.
So grammar's spelling, like a year on each of those would be really miserable. But in some other sense, couldn't they be helped by an extra year of training before coming to these books? Does that line of questioning make sense? They're unprepared in some way.
They could be better prepared. And I think that's why math is structured the way it is. We'll get to that eventually. But does that make sense as a question?
So you would say that we shouldn't do things that way, but they should dive into books before they're ready for them. Is that also correct? I don't know if I would say that. I think that you were disconnected.
Training in the spelling and grammar is for the composition, not for consumption. So let's say in your class you do also have essay writing. Maybe this is a better example. You guys say in your class, even though you're asking ninth graders to compose essays on huge topics that have been debated for thousands of years in written history and then before that in oral history.
In that sense, they are absolutely unprepared to opine on certain things. But you still have essays in your class. And I would much prefer that they spent an extra year training in grammar and spelling. Before writing more essays.
So then would you ever consider cutting essays from your curriculum to then have Graham better prepared to assign those essays? No. I mean, we're talking about the way even our school is structured and the way student development is structured. The brain of a small child is particularly tuned for grammar.
Like they like to collect facts and they like to memorize things. And so the grammar needs to happen when they're young. It's incredibly difficult to teach at my stage. I will teach them things and it will just fall right out of the brain.
And it's just not prone to it. And then the systematic organization of language should happen probably somewhere in your junior high when their brains are trying to make connections and figure out logical things. In high school, those kids just want to be known. That's the time for expression.
So by the time they meet me, they should be prepared with spelling grammar and structure. That's the ideal. We have a hard time meeting it because we get kids from different schools and changing curriculums in the younger years. But that's the ideal.
But even at those younger grades, they're still writing essays. Well, yeah, your saying they should be, expressions should be a part of each grade. But at younger ages, they should be focused on like spell the words correctly. And then they should be focused on make your logic coherent.
Make your sentences actually structurally sound and coherent. Make sure each one makes sense. When they hit essays, that's when they hit me. Grammy were shaking your head at points along the way.
No, no, I agree. For spelling little kids, elementary school-age kids, love memorizing those little songs. And the best songs for them to memorize are like rules about grammar, i, before e, except after c, things like that. Above, beyond, below, beneath, whatever the song is for all the prepositions, right?
I don't know the song. I remember the palasas though. So some of our kids have this song that like, ba-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da-da. I remember I had other prepositions.
Little kids love doing that stuff. They don't know why they're doing that stuff. They're just singing a song. And then in the middle years, when kids start being jerks, not jerks, being precocious and saying, why?
Why do I have to do this? Why is this guy blue? Why do I have to do homework? Then they're wanting the connections to be made.
And so that's when logic is taught. And that is when arguments are made. And kids will want to, if someone says, you can motivate kids to write things by saying, I'm trying to give an example. I will only answer your question if you can write it out in a good sentence.
And he'll be like, fine. And then he sits down and writes it out. And then you can point out, say, no, you didn't do it correct. And he'll be like, right, you do it over.
And then because they're motivated to get to that answer. And so by the time they're in high school, yes, AJ is right in saying that something in the development of the hearts and minds and souls of a kid is that the only way they want to be able to express and be known. And they want to be heard and understood. And if they don't have those tools of grammar and logic, then they're wanting to be heard and understood, manifest itself in ways that they will make themselves further understood regardless of what skills they have with tools.
And they don't have the skills with tools. They'll be heard and understood in ways that don't require them and are perhaps sort of less refined. So I guess to bring the analogy back to math, little kids should be memorizing equations. They should be memorizing A squared plus B squared.
Yeah. This is great. So this is not the direction I thought that we'd go, which again is like the title of all my episodes. But so I didn't expect to trip him direction.
But I think you're onto something here. So at the point that when you all were in high school, or even you teach high schoolers, at the point that they're in high school, they're still doing grammar work in math. So here example with the Pythagorean theorem, a thing that they need to memorize is A squared plus B squared equals B squared, or the quadratic formula. That's a thing that must be memorized.
Yeah. What is the rhetoric of math, I guess, is the question. That's one way of getting at it. That's one way of getting at it.
What's another way to say it? That math is like the Pythagorean theorem. What is the purpose of the Pythagorean theorem? Exactly.
The volume of a triangle, right? Isn't that the Pythagorean theorem? I'm thinking the, no, that's not the right way. It's length of the hypotenuse.
Yeah. Yeah. For a right square, if you have the length of two legs, you can figure out the third leg. But there's something like bloodless of just presenting, this is the formula, go forth and memorize it.
Yeah, no student, when we're reading Paradise Lost and we're talking about Satan's soul and despair, and we're having this as a conversation just to think back to the last week's podcast, no students are like, Mr. Donaldson, when are we ever going to think about souls and despair in our life? Right? That doesn't happen because they can see why that's a connection.
Now, English is sort of easier. It doesn't easier bridged across in English class than in math class. But isn't there would be a bad way of teaching English where you teach the themes ahead of time, and you're like, memorize these themes. And then recognize them in the story.
And give the homework assignment that's on page 433 of Paradise Lost, Satan, experiences despair. What are the three, that would be a bad way to do it. I think it's closer to most people's typical math experience. And again, this is not, I'm not commenting on this rule.
I'm just saying, like, math can be kind of boring for people that aren't interested in it, they've accepted. Math should be, I think, I've never taught in math class. So this is definitely not even an amateur comment. But I don't see why it can't also be done in the same way, in a question-based way.
I mean, Aristotle, Socrates does this with the slave boy. And what is that story? He's trying to prove that this slave boy has, like, precognition knowledge of it. And if I'm not mistaken, he is able to talk this boy into the Pythagorean theorem by asking, again, it's a pragmatic method.
You ask questions. But then to get to the point of drawing out this triangle and realizing this relationship between the three sides. Yeah. So in my example of the murder circle, what we do is we start to put a table together, and we say, all right, if there's two people, where should you sit?
Number one. If there's three people, where should you sit? One kills two, three kills one, sit in three. The next one, if there's four people, you should sit in one.
Right? And then they go through it, and they realize, like, wait a minute, it goes one, three, one, three, five, one, three, five, seven, and then they see that it begins to cycle and repeat. And then they see a pattern, and then they try to figure out why there's that pattern. And so I don't teach them anything.
We're just going through questions. In my mind, I know what we're trying to get to. But in the student's mind, they're just trying to figure out how not to get murdered, which is a great motivator. Yes.
So I feel like, yes, I don't know if this is what he gets in that in his essay there. But that is a way to do math. Now, is that a way to do pre-cal? I don't know.
What a fast way to do pre-cal, is what about it? It is certainly not a fast way to do pre-cal. We'll get into his end of the essay is just his take down of every grade of math, algebra 1, geometry, algebra 2, pre-cal calculus. But even that conception of them being in different categories doesn't really make sense.
It's all math. It's all a logical approach to numbers and shapes, just let them into algebra geometry. It may not make sense or be helpful. I guess, tying in somewhat with your just even example, he's talking about how to teach the area of a triangle formula.
This is base times height divided by 2. So there's one way of teaching math where you put that on the board and say, if you want to figure out the area, how much space is inside of this triangle, here's b times h divided by 2. That's how you would present that. But if you were to try and figure that out, that's like, you would never phrase it that way in the first place.
You would know what a base is. You wouldn't know what a height is. And why isn't it enough to draw the similarity? If you take base times height and multiply them together, what shape do you get?
What area of shape do you get? A square, triangle, right? Or a triangle, square or a rectangle, right? It would work for either one.
Well, then why not take a rectangle, draw a triangle inside of it, and then the problem for the student is, what's the relationship between these two things? Let them figure out along the way, oh, the rectangle is the full amount, then the triangle must have half of that area inside of it. Or say like, draw a line and say, okay, how long is this line? They say it's four inches.
Or do you do inches in math? We did centimeters. I mean, we did inches for sure, but four inches. It probably should be, you just say units.
It's actually a matter of guess. So four whatever. And then we say, okay, we're going to stack four whatever's on top of each other up to a certain height. And so how many of these should we do?
15. All right, so we've got four, and then how many of these are on top of each other? 15, okay. And so then the area is four stacked on top of each other by 15, and so then they find out that four times 15 is the area and we've created a rectangle, right?
Like, you know, there's ways to do it. It's just, you know, anyway. Okay, that all sounds real grand, but. We've got an economy.
And those three problems are. Well, I mean, we kind of do have an economy, right? So here's the point, is that mathematics is not something that you can easily grasp in what, the seven years between ages five and 12, and then 12th grade? You know, we'd out the dimensions are.
No, my point is, is that some of these things, some of these principles and formula were discovered by somebody after years and years, or even the lifetime of work, and the proofs for them are incredibly complicated. How are you going to get your students to do that over the course of an hour? And the expression of math, say, we can easily get to an expression of math in small cases, right? With geometry, I can make a rose window, I can do some of these basic things, but it starts getting more and more complicated really quickly.
And if the thing that they eventually need to do is figure out the weight, like how much each joint on a bridge will bear weight as trucks go across at a certain speed, and wind is coming from sheer direction, and the gravity changes because of the tide and the moon. Like, my point is that when they hit senior year in high school, they might still be in the grammar stage. They might still be gathering the grammar of math, and the problem is that we don't get them to the expression of math because the expression of math is so complicated and it comes in college, right? And math has been developing and rolling over years and years, it would be like trying to teach science, and then trying to have us establish and get them to the point where they discover osmosis and discover how to calculate mass and discover.
Like those, all of those things can be really complicated to get to, and we simply don't have enough time to get to the place where they need to be via the question method. That has always been my beef with this guy's notion. Yeah, it sucks, it sucks to sit and learn grammar way past when your brain wants to do it, but if you ever wanna get to the actual expression, you kinda have to at some point. Do you think people get to the point of expression as is now?
I think that more expression, and what I mean by expression is like using math to do things, right? So using the, figuring out the hypotenuse to figure out how much cloth you need to dissect a room by putting a curtain, right? I think these things could be done, but you can't do it with everything. And you just, you can't say it safe for learning Latin, right?
Everyone hates the first year of Latin, you have to memorize words, it sucks, it's no fun. But the joy of Latin comes when you get to read poetry and it's original and you get to go to Rome and read the stuff that nobody else can read. This is great, no, I think, I mean, I don't know what, I don't know what, I don't know what I'm saying. Yes, but he would argue that that never happens in math.
So you start with algebra, and then you don't, it's not like then in geometry, you move to expression, you then you go back to grammar essentially, again, I wasn't thinking of this from the Trivia Mangle that you two are taking it, but it's true, you restart every year with a totally new field of study essentially, as opposed to 10th grade English, you're better off for if you paid attention in ninth grade English, right? In a way that geometry is actually pretty different than algebra, yes, they both have numbers, but a plane versus equations are very different. I don't know. But many of them do end up in the expression of grammar, just doesn't have any high school, they go on to business where they have to use, they have to calculate profit and that to figure out, okay, we're gonna put my stop losses if I want this particular goal over this amount of time.
And then if they're an engineer, it's even more prevalent. They have to figure out how to cut down a wind resistance on a thing or create a canoe that has a certain amount of displacement because it's made of concrete, right? And they have to get concrete to float because it's canoe. I've seen these projects in engineering schools, we still call it math anymore, we call it engineering, but that's the expression of math.
And we call it architecture, whether you're doing math or we call it computer science, where they're doing both logic and math. Let me take your example. It would be equivalent, I think, to the earlier mode of teaching English by only degree or only do spelling, learn all of your themes before you actually read a book. As opposed to take what you just said and move it earlier, why can't you start, I don't think real world application is necessarily the point of math, even physics you often will assume like an environment with no gravity or no friction, you'll do not real world examples, but even take the, why can't those be the subject of what you're learning earlier on, as opposed to, I have to do all the prereqs beforehand and then I get to real math in college.
That's even the music example. I went all my music theory and then I pick up a violin. And I think that's what he would say is the problem. As opposed to math is not a series of things that are solved, it's a field that is still expanding.
There are the, I forget what they're called, but they're like a set of mathematical questions that are unanswered and there are a million dollar prizes for answering each one, I'll try and look it up in just a second. But why not present those first and foremost? And then we're looking at how to, how would we get at answering those questions as opposed to memorize all these formulas and not explore what relationships you can find in a triangle and appear in a picture. The thing that gives me pause is that the system that we have set up, the kids that go off and do pure math or who are interested in solving those problems and whatnot usually end up getting plucked out of the system to go do it properly.
And so you have the OR, they get punted up to the older grades because they can do it properly. And so we say, all right, here is a sixth grader who actually is really good at this. And so we boot him up to 10th grade math or whatever. You see the same thing in music programs.
Oh my goodness, this kid who's in band class is actually really great, we're actually gonna get plucked in band class and go to youth orchestra in the city. And so there's a sort of point, we don't have something equivalent in English or history or the humanities because you can kind of like, I don't know, stunt them or whatever. Although there probably would be enough to creative writing sections. Like if we forced them to write short stories every year and the kids who really wanted to do that would probably get plucked away and go and do it for real.
And even saying that they're going and doing it for real, they're gonna leave our math school and go to quote math for real somewhere else. What is the real thing about? Is like this is true of all schools. And so that to me is like an indictment on the way that we've done it.
So is that because we say every human person should hit some sort of competency of math. And so that as time has gone on, we've kind of created a system like the kind of math that I did and we all kind of did the system of compliance of sort of doing the systems that are presented to us and memorizing this stuff. And the kids who actually like kind of get behind the curtain and are like, I like this. We're like, all right, we're gonna take you and send you to like where people are doing this in actuality as opposed to our neutered version of it or whatever.
Sure. Graham, AJ, before we go any further, I want to thank our Patreon sponsors for making this episode possible. Our Patreon sponsors support us at one of four levels under the go through right now. I think many people listening, they want to be part of this as well.
They want to become patrons as well. We have a $2 a month tier. Those are Ghibliens at $2 a month. You get access to all of our episodes ad-free.
You also get access to previous content that we've done mostly at conferences. So you get access to many other bonus episodes as well. At $10 a month, you get access to our in-between episodes, which we record after every single episode that we record. You also get access to our monthly AMAs, which I think are really funny.
Some of our best content. In addition to all the same benefits at the $2 a month tier, you get access to ad-free episodes. Above that, at the $20 a month tier, at that point, arguing input into the podcast, you are helping us come up with future topics to come up with future merchandise, in addition to all benefits from the tiers below that. And finally, you heard about this in recent episodes.
We have added a Heliosus Acolytes of Love tier at $100 a month. At this level, you are a true believer, and you are the most faithful of our listeners. At this tier, you get all benefits from lower tiers. You also get, I can't believe I'm saying these words, that you get a Heliosus Acolytes of Love crew next sweatshirt.
You get Heliosus Acolytes of Love crocs, and you get a free copy of all future merchandise as we create it. Incredible, incredible benefits at this level. That is only for $100 a month. You can find all of this at patreon.com slash classical stuff.
Thanks again to our patrons, and thank you all for listening. Sure. And Lockhart would argue that sometimes what will happen is that the people who are good aren't good at math per se, they're good at memorizing, or they're good at following instructions, and then when plucked out, find out they're not actually good at math. You know, it's the same as- Oh sure, yeah, yeah, yeah.
I get to college and they do some high level math and start doing a little bit of stuff. I'm not interested because- Just tell me the answer so I can memorize it. Because we're good at solving problems, not thinking about these things critically. So this worries me a little bit because we- Or it makes me think that maybe we should have kids start specializing younger.
And then the other side of me says, no, we should have well-rounded students that love all of these various classical subjects of math, English, history, philosophy, music, art. And they should be exposed to all of these things until they're 18 and leave school. Sure, but I mean, I'm sure they're thinking the same thing. There are no subjects, right?
At some level, even to say, yes, students should have more math, but they should have more math because art is mathematical, because science is mathematical. And even, again, I only ever feel like I say the same three things over and over again, a classical approach to math would probably be closer to history and a history of mathematics as opposed to here learning the formula. I don't know. So that's in, yeah, I don't know how to- Just same thing with science class.
We talked about science class. We talked about science class would be a history of biology, a history of scientific thought. Yes, and that's how you would get through in the same way that what, yeah, does this get to anything, AJ, for what you were saying before? Obviously, we cover thousands of years of history and we'll never do justice to all the amount of time that we're covering, but still we say that we offer history class, right?
I'm trying to be charitable to the people who do this as a profession and really care about it. So to speak to Graham's issue, that everybody has to reach a certain level of competency. So say I live in a society where everyone is eventually going to have to make their own knife because we don't sell them, right? We don't sell knives.
Everybody's got to know how to blacksmith a little bit because I'm going to do your knife and for you, right? Or you're going to have to pay me to do your knife and that's going to cost a lot of money. So everyone should learn how to do it a little bit. Well, a few people that really are like, I like making knives.