Episode 95: A Mathematician's Lament. A Conversation with Paul Lockhart.

We talk with mathematician and author Paul Lockhart about how school can drain the life out of math and why real mathematics feels more like art than a subject. We argue for desire, honesty, and beautiful explanations as the center of learning, then share concrete puzzles that show what math looks like when it is alive.

• why “school teaches school” and how schoolification ruins natural curiosity
• learning as a personal gift and teaching as inspiration rather than control
• mathematics as pattern, proof, and the art of explanation
• building a mathematical worldview through questions and play
• practical ways adults avoid killing desire in kids
• what a workshop-style math class looks like without tests or homework
• domino tiling and checkerboard reasoning as a model for elegant proof
• why formulas stick only after you can see the idea
• protecting your mind and reclaiming happiness as a learning goal
• a surprising shout-out to a student who changed Paul’s thinking

About our guest
Paul Lockhart is the author of Arithmetic, Measurement,  A Mathematician’s Lament, and Mending Broken Bones. He has taught mathematics at Brown University, University of California, Santa Cruz, and to K-12 level students at St. Ann’s School in Brooklyn, New York.

Connect with Paul
A Mathematician’s Lament
Arithmetic
Measurement
Mending of Broken Bones

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Transcript

Schoolification And Lost Learning

SPEAKER_02 0:00

School really teaches only one thing, which is school. You can schoolify things. You can take something that we love to do as people, socially, individually, and you can ruin it. You can make it mandatory. You can make it you're sitting in desks facing in one direction on a linoleum floor, but it isn't anybody learning anything. It's a complete waste of time. I mean, okay, look, I don't know. I'm not here to say this is good, this is bad. That's your decision. And that's in fact my whole point.

SPEAKER_04 0:32

Hello, and welcome back to an exciting episode of the Kid Look Podcast. I'm Kelly Smith, and I'll be hosting today interviewing Paul Lockhart. Paul's a mathematician. He spent most of his career as an academic mathematician studying things, working on new knowledge in the field of mathematics, and eventually left academia to spend time helping kids encounter math and immediately figured out that what they were calling math in math class was often not mathematics. He wrote a book called A Mathematician's Lament back in 2009 that really just outlines in big ways what's wrong with education the way we talk about math. So you're gonna love this conversation with Paul. He brings such a fresh perspective and insights into all of this. Paul's also the author of a book called Measurement, a book called Arithmetic, and a recent book called The Mending of Broken Bones. So I hope you're able to enjoy this conversation with Paul Lockhart and rethink the way you consider mathematics in your own life. Okay, Paul Lockhart, welcome to the Prend a Podcast. We're gonna be talking about kindling today.

SPEAKER_02 1:39

Terrific. Well, thanks, Kelly. I'm glad to be here.

The Lament And Why He Wrote

SPEAKER_04 1:42

I got to know you through a book you wrote. I'm sure a lot of people got to know you through a book you wrote. You've now written multiple books. You've been a math teacher for a long time, but I think what stood out to me about this is, well, so much more than a math teacher will get into. Your book's called A Mathematician's Lament. Lament is a pretty strong word. I think some people listening will have read it. I'm gonna put a link to it in case anybody hasn't read it. Please read this book. It's very useful to just think about things. But can you just give people in a nutshell? I mean, you start out with this musician having a nightmare and just tell people what your what your point is in this book, because I think it's important for them to understand.

Learning As A Gift To Yourself

SPEAKER_02 2:20

Well, the book has a very, it's a funny little book. It has a funny origin story. I had left academia and was teaching at a private K through 12 school in Brooklyn, known as St. Ann's School. And I was getting to be pretty good friends with the founding headmaster of the school, Stanley Bosworth. And uh around 2002 or 3, he started talking to me about a book project he wanted to do, a book about the school. And his philosophy was your teacher should be someone who does that thing. You know, your art teacher should be an artist, your music teacher should be a working musician. And it's true that, for example, the music department at St. Anne's, the teachers, you know, teach music by day and then play on Broadway and at the Metropolitan Opera by night. You know, they're working musicians in the city. So they they have a point of view. And, you know, it's pretty hard to get a working professional mathematician to teach at your elementary school, but I was an unusual person in that I really love kids. And I started teaching kids when I was about 18, actually. At a I did an afterschool program at a at an elementary school in my hometown. And I always, you know, as I was teaching at universities and working with grad students and all that kind of stuff, I, you know, I missed it. And so when I decided I was done with the academic world, I really wanted to go back to working with kids. And so anyway, it was a it was a nice alignment. And and Stanley wanted me to write the math chapter, you know, about the school's philosophy, what we do here with math. And so I said, All right, that sounds fun. And I'd never written anything really before. I'm not a writer, you know. I'm written math papers, but and then something funny happened. So that summer when I sat down to write this chapter, what ended up happening was all my pent-up frustration, anger, bemusement, all my feelings about math and school just kind of vomited out onto the page in the form of this diatribe, I guess you could call it. In any case, it's a it's it's an opinion piece. And, you know, it's basically my view that school, as we've come to know it, does not really serve the students, the teachers, or anybody else in terms of any kind of legitimate education project. School is what we all know it to be. Boring, a place where you're ordered around, you're sitting in a desk. How's that learning? Well, it isn't. And so that's what I wrote about. And of course, you know, my expertise is in the area of mathematics and math teaching, so that's where I, you know, bring my arguments to bear. But I think pretty much anyone who's interested in anything could say the same thing about what school does to their subject, because school really teaches only one thing, which is school. You can schoolify things. You can take something natural to the human animal, you can take something that we love to do as people, socially, individually, and you can ruin it. You can make it mandatory, you can make it you're sitting in desks facing in one direction on a linoleum floor, listening to some human uh drone on and on and enjoy their little authority uh game, but it isn't anybody learning anything. It's a complete waste of time. I mean, okay, look, I don't know. Maybe you loved school, maybe you had a great time. Okay, fine. I can only speak about my experience. Yeah. Not here to say, this is good, this is bad. That's your decision. And that's in fact my whole point.

SPEAKER_03 6:07

Right.

SPEAKER_02 6:07

I would like to redefine what we mean by thinking, learning, and teaching. Because what school does is it puts you in the position of having to accept this model as what learning is, but it isn't. And so for me, like to I'd like to sort of position my understanding of those ideas in this way. Learning is a gift you give yourself. It's part of your 80 or 90 year project here on earth, trying to understand, pursue happiness, enjoy your life, all that kind of thing. So learning is individualistic. It's your own search for meaning and beauty, etc. It's not for anyone else to do to you or tell you what that means. Teaching is the inspiration of learning. It's someone who's an active learner who loves to learn, showing you live their excitement, their passion, who they are, being honest, opening up a world for you. And then you can decide what you want to do that. You can pursue your artistic agenda, your historical agenda, your mathematical agenda. You can do your thing because you've seen what it is. So that's the kind of thing that I want to accomplish if I'm in a situation where I've got some students. They're not students, but if they're not seeking something. My friend Harold used to talk about the student-to-pupil ratio as being the important thing. That some people are here because they actually want something. Others are going through a bureaucratic credentialization process and they don't care. And so we have a we have this really kind of it's both tragic and comic. We have a tragic comic situation of people being in a room, students who don't want to be there, teachers who don't want to be there, and you kind of almost, maybe even openly, there's a there's a admission that we're all stuck here because of the government, and we're all just gonna somehow have to get through this thing. Well, that's sad. That's so terribly sad compared to what it could be. So a mathematician's lament is in two parts. The initial thing that I wrote, the the lamentation, is just me expressing my sadness, my anger, hopefully in a reasonably fun and enjoyable way, about what's been done, my art form, the mud it's been dragged through, the awful impression of it that kids are given. You know what's funny? This is a funny thing. If you're a painter and you're on an airplane, and the person next to you says, What do you do? and you say, I'm a painter, well, then you might end up having a really interesting discussion about art. If the person says, What do you do? and you say, I'm a historian, say, Oh, really? Oh, I've ever really been interested in the Roman Empire or the Civil War, and you're talking about history. If you say you're a mathematician, you're going to be talking about school. And math doesn't happen in school. That's not, there are mathematicians. They do math. That's what they do. That's what I do every single day of my waking life. Has nothing whatever to do with school. But culturally, we don't know what math is. It's just that thing that you do in school and you try to get out of as quickly as possible.

SPEAKER_04 9:44

Can we talk a little bit about this, Paul? Because I think this is so insightful. And by the way, I'm shocked to hear that you came straight out of academia. It felt to me reading this like you had been in every single one of my math classes growing up.

SPEAKER_02 9:59

Well, that's because I went to school, you know.

Math As Pattern And Beautiful Proof

SPEAKER_04 10:02

And maybe it just hasn't changed. Yeah. I mean, I was a kid that did well by the standards, and I thought I was good at math. I found out later that I was good at following rules, as you point out in the book, right? I was good at following instructions and working my way through. And really, you know, I didn't even know what math was. Like, let's talk about what math is, because I'm I'm look reading this out of your book. The mathematician's art, asking simple, elegant questions about our imaginary creations and crafting satisfying and beautiful explanations. I mean, that sounds fun, right?

SPEAKER_02 10:34

That's something it is really fun. I mean, you know, any artist is going to tell you about their art that it is fun, challenging, impossibly hard, that the greatest joys in your life come from it, and the deepest pain in your life also comes from it. So, you know, we we should understand that writing, poetry, sculpture, these are all forms of suffering on some level, but they're also forms just like giving birth contains a fair amount of suffering. You know, anything that you're going to bring into being and subject to your own personal aesthetic criteria is going to sometimes charm you, sometimes pain you, sometimes embarrass you, sometimes thrill you, all of those things. So it is just, it's it's a craft like any of those others. And, you know, math has been called by mathematicians, it has been called the art of explanation. It has been called the music of reason. Mathematics really is an alternate universe of pure pattern. It is our the place we go when we want to distill the beauty, simplicity, and elegance of pattern. That's where we discover it. So the work that you do as a mathematician is partly discovery. You might hit on a new thing no one's ever noticed before, a very interesting pattern that's happening with shapes or numbers or arrangements. So that's really interesting and exciting. Another aspect of the work is you want to understand why this pattern is happening. It's one thing to notice a pattern, it's another thing to say, what's going on here? Why is this happening the way it's happening? And so what we're after is explanations. And that's what we're interested in in our mathematical arguments. Also, we want them to be correct, logical. They want they should make sense and all that. But maybe even more important is we want them to be beautiful, charming, elegant. We want them to make us gasp. So um the second half of a mathematician's lament is the exaltation half, and it's about how math is so beautiful and what it's like to do it, and how it makes you feel. And so I think um it's a good book for people to read. If you've always wondered what is this thing called mathematics, surely it's not sitting in a desk while someone forces you to move symbols around on a piece of paper in in a way that doesn't mean anything to you. That can't be what all these thousands of really interesting, smart, creative people have devoted their lives to for centuries. So I hope that people who read the book enjoy both halves. You know, that they can identify with the part about how the school experience is so ugly and so beneath our dignity, really, as humans. I think in my darker moments, I think I tend to think of school as a kind of ritualized societal mental child abuse. Not that the adults aren't also being abused mentally. Just the whole school experience is mundane. It's such a tragedy to take things that are in and of themselves so human, so warm, so feeling, so pretty, so fun, and to schoolify them, to make them to put them in the service of the corporate capitalist kind of wage slavery agenda. You will have a manager he will order you to perform labor that you don't care about, get paperwork in on time, and watch the clock, so that's what you do. And I'm sorry, maybe maybe some people like that. And if so, great, more power to them. Enjoy your school and workplace experience. I did not enjoy it. I did not enjoy one second of it, and I was in trouble all the time. I was a terrible student. Got kicked out of schools when I was really little, because I refused to take orders. Pretty much just kind of tuned out in the sort of middle school years, and by high school it was active rebellion. I don't know if I officially graduated from high school. I'm not quite sure. My sister definitely dropped out. I bailed in a few weeks into college. Like, no, I got math I want to do, and this is a waste of my time. So I don't have a solution. I'm not sure that there really is a top-down solution. I think things like governments, churches, the military, school, these are these are just these beasts are too huge. So, what can we do? What can you do as a student? What can you do as a teacher? Rebel. Of course. Rebel. The extent to which, the style of which, the nature of which, you're gonna have to figure out what you can get away with. I think most teachers would say, I fly under the radar, I try to meet my students halfway, we try to do something worth our time while closing the door and hoping the administrators don't walk by.

Questions That Train A Math Eye

SPEAKER_04 15:56

Well, there's lots, lots of structural things we could talk about there. And of course, you know, I our listeners will know that I spend my career trying to help people start alternative forms of school where questions, for example, are good, right? That's that's one thing you talk about in your book that's just the art of a question. I hope uh, you know, our our listeners there's some of them are on audio, but some of them are on YouTube. So I want to do this together if this is okay. This is a picture. This is a picture from your book, okay? So you take you take a rectangle, you say, all right, here's this rectangle. I wonder how much of that rectangle is taken up by the triangle inside the rectangle, right? I think that's just a brilliant question. And I guess basically what I want to do is is sort of push on that. Like, where does a question like that come from? And how does that influence the minds of young people? Because you could do that even inside of a tightly confined or constrained public school setting.

SPEAKER_02 16:48

Here's maybe the a way to start thinking about it. If you're an artist, I think you have a worldview. A worldview that means that if you're walking on the beach, you're gonna be picking up on things that maybe other people aren't picking up on because you're a painter. And you will get ideas from that walk on the beach. And if you're a musician and you're out in the woods and the birds are tweeting and the leaves are rustling, you have a different relationship to sound than other people. And so your worldview frames your experience out in nature, and it might give you ideas for a symphony. There is a mathematical worldview. It is being sensitive to pattern, a sensitivity to pattern that is so highly refined, it's very hard to even get across. No, it would take a good, you know, you and I could spend days or weeks or have a kind of a workshop together to get across what that is. I'll try to say some things, you know, to help get that across in a second. But because you have that worldview, all life experience is just for that mill.

SPEAKER_04 17:58

Before we dive into what that worldview, is that worldview fixed? I mean, are there certain people that are more inclined to asking questions about patterns? Or could it be instilled or could it be developed for lots of people?

SPEAKER_02 18:13

I think it's natural. I think children are all musicians, painters, and mathematicians. Okay. Until something gets damaged. See school. No, I don't think. Okay, look, I mean, is everyone Mozart? No. That was a very unusual three-year-old. Really unusual. With a violin teacher dad who was extremely aggressive about it. So that could have gone very, that could have gone south in a big way. Right. Happened to work out. And no, I don't think that everyone is gaussed. You know, some people have more of a desire for that. I think desire is really the issue. Desire, the thing that you don't ever find in school. Desire is the heart of it. I, as a child, I played with beads, string beads, lots of kids' string beads. My idea of stringing beads, and I remember this day very particular, I was probably six, maybe seven, where I had this box of colorful plastic beads, and I had, you know, long shoelaces or whatever. And I hit on the idea of doing every two-color alternation pattern. I'm going to make them all. I had maybe 12 colors, so I didn't know how many patterns that was going to be, and it got to be a lot. Then I noticed things like some two-color alternation patterns resonate. Like blue and yellow says Sweden, right? In the same way IKEA does. And green and red say Christmas, loud and clear. Black and yellow is honeybee. But really, it was having the set. I became a stamp collector later as a kid, too, for a similar reason. It's the feeling of having the pattern, understanding it. Yeah. You know? So I think that all people are mathematically inclined. A lot of people are ma are do math and don't even know it. Like a Rubik's Cube? Can you tell me a more popular fad in this entire world than that? That's a piece of math. Sure. A really, really beautiful problem in modern algebra introduced in a gorgeous way. There's beauty just dripping off of that object. There's mechanical beauty, there's mathematical beauty. It's gorgeous. Sudoku are math problems. There's a very funny thing about Sudoku, by the way, which is when you look at a Sudoku puzzle in the in the newspaper, they always have this little disclaimer. Don't worry, there's no math. I've seen that. What they mean, of course, is that you won't be doing arithmetic. Yeah. Because these digit symbols that you're writing in aren't numbers. They're just distinguishable symbols. They could be letters, they could be colors. It doesn't make any difference. Right. But nothing's more mathematical than a puzzle like that. You're trying to arrange all of these different colors or letters or whatever so that there's one in each row and in each column so that this other condition, that's a beautiful little problem in combinatorial geometry. It's a interesting. It's a solid, legitimate piece of mathematics. Okay, so we've lots of things like that.

SPEAKER_04 21:24

We've got parents and educators listening, people running their own little microschools, and they they're free. These people do not have all the structures and overhead, and there's no chance of an administrator walking in.

SPEAKER_02 21:34

I'm so happy for you people.

SPEAKER_04 21:36

So here's what I want you to tell them is everybody starts out asking these questions and looking at these patterns in the world. What can the adults in their lives do to not kill it? I mean, I mean this question sincerely. Like, tell these people, because I think what we would love to do is help more people feel both more confident being a musician and an artist and a mathematician, like looking for this wonder. in the world.

Teach Honestly And Drop The Theater

SPEAKER_02 22:01

Well, you know, the the teacher role, it's a it's a kind of a harsh position to put someone in. Because you're kind of having to live up to the idea of you being some sort of an authority figure. You know, it could be a disciplinary authority figure. Yeah, I I don't want to be that. Sure. That's not what I signed up for. It's, you know, you having to pretend you know something about this. Well just be honest. Be honest. That's my biggest advice to anyone who's in the position of being a teacher. Be as honest as you possibly can about what you believe in and what you enjoy and what you think of things and how you feel. Be honest about your level of ignorance. There's nothing wrong with saying, you know, I'm not a person who does this. I don't really know that much about it. I can help you to the extent that I can. Be a person who is not trying to cover material or push information from you onto other people. Be a learner. Be a person who's trying to uncover. Be a person who as a learner recognizes other learners and appreciate the fact that each of us has our own agenda and our own criteria for when we feel we've understood as much as we want to right now. So I'll tell you you know this is going to be, you know, what I'll tell you what I did. Okay. I I was in a school and this is a, you know, this is a school school. This isn't some experimental, you know, mom and two kids, you know, on a farm trying to do something awesome and alternative. This is regular old school. And I to this day don't do not know how I possibly got away with this. It would be interesting to have some of the administrators who slammed their doors in my face and screamed at me on the program to answer this question. How the hell for 20 years did I get away with running my own little school? So I could not bear to do school. I couldn't do it. I couldn't do it when I was a kid. I'm not going to do it as an adult. So what what's sensible here? What makes sense if we're going to do mathematics together? Workshop. That's what makes sense. So I never took attendance. I never gave a test. I never assigned any homework. I never evaluated anybody. I never assessed anything. I only engaged in mathematics with my students and we had an absolute blast. They learned a ton. They had a great time. My little compromise was at the end of the year they write me a letter describing their experiences and that's what I hand to the administrators. So you know I'm not recommending this for teachers around the world who are working in public schools. You will get fired if you try to do that. I don't know why I wasn't. One thing that I guess worked for me was this is a private school of very wealthy New York City parents who really want their kids to get into Ivy League schools and I'm an Ivy League professor and their kids get to work with me now. So I was very popular among a certain group of parents and a certain group of students and extraordinarily unpopular with another group.

SPEAKER_04 25:34

Sure. Did the kids grow up to love math? I mean did you see differences in the outcome that way?

SPEAKER_02 25:41

Well you know I was open to anyone who wanted to engage on whatever level they wanted to. And so that included the absolutely most precocious math students in the school of course and many of them went on to grad school and you know beyond and published papers and are mathematicians and sure. But I also was very much of a favorite teacher for the kids who were really, really bad at what you would say would be the kind of traditional notion of math, you know, doing arithmetic on command, things like that. Quadratic formula because um I was talking about the real deal which is exciting and interesting to every human. And I wasn't demanding that they achieve some level of technical fluency or anything. I was just saying weigh in what's your idea? What do you think? Let's do this together. So of unlikely that those people are now working professional mathematicians, but very likely that they are adults in their life who have an idea of what mathematics is and why some people devote their entire lives to it.

SPEAKER_04 26:49

Yeah. Well I'm just saying it, you know, you you've made this point in in multiple ways in the book but and and I know you've done this since with measurement and some of the other work you've done. I I was asked to read Hamlet and Macbeth, right? So but I don't think it was because I'm predicted to be next in line for a kingship of Scotland and that I should resist murdering the king so that I take over his spot. I mean if you were to ask you know those teachers back then what does this have to do with my life it's like it's interesting. It like tells you something about yourself and the world you live in in a in a very abstract way. It's it's decoupled and and I think too often with math and you've you've made this point that it's it's really trying to be shoehorned into this is so that you can invent the next self-driving car or something but which does need math to do that. But like there's some there's a better reason to learn math. And I think you hit the nail on the head with the way you did it as a teacher and I'm I'm so grateful you got away with it. Even more grateful you wrote about it because I think that's super helpful.

A Classroom Without Tests Or Homework

Arithmetic As Meaning Not Algorithms

SPEAKER_02 27:49

Yeah and actually my other books came out of that experience too. So one of my friends at the school was a first grade teacher and she confessed honestly and you know with some anxiety and some trepidation she confessed to me that as an arithmetic teacher for six year olds she really felt uncomfortable. This isn't something that she really understands very well. I mean she can follow a rote mechanical algorithm just like the kids she's training, but it doesn't mean anything to her. And she the things that she does with the kids apart from that were things that she wanted to do that meant some to her. She was really into ancient Greece and ancient Greek art and the kids were designing temples and learning about metopes and Ionian columns because that was her thing that she loved. And so I agreed to come visit her class a couple times a week and do arithmetic with the kids. Like I became the arithmetic teacher that was a was a visitor. And the whole thing was a total hoot because I'm just this fun guy who comes in. I'm not putting them under any pressure, you know, judging them in any way we're just talking about arithmetic. You know, the night before I was going to visit them for the first time I was kind of thinking what the hell is arithmetic and and what do I want to say about it? And so that's what what I chose to do was to do what is honestly the truth. Start from from the way it actually started 2000 years ago with early humans, objects, language. And so out of that experience came this book Arithmetic that I had a great time writing and lots of people have written to me saying it changed their life when it came with that because you know parents don't really understand arithmetic. A lot of teachers don't and it's just being sort of do this then do that then carry this and write this here and then shift these over why that's how you do it. And it's like oh that's horrible. Let me actually while we're on that subject let's let's have a little example. You were saying you know what does math look like now the problem is if you said you know what is what is painting? We don't know anything about painting over here. Painting is the thing you do to the deck furniture to protect it from the weather. What's this representational art stuff you guys are talking about? What is this so-called beauty with a capital B? Well you know you'd be a little nervous about showing any one particular piece of art because you know it it's supposed to stand for everything. So you'd want to do a few, you know? And maybe you don't start with you know the Jupiter symphony or Michelangelo or something. Maybe you start with a cave painting. Because you can really appreciate that hand on the Lasco cave that was put in the ashes and that's I exist. So maybe we start with something really small. So let me give you a couple of examples. This is not high art. This is not the art of the professional working mathematician but it in it's it's a small version. It sort of gets across that what we're going for and what we mean by elegance. Maybe maybe you'll tell me how you react to this. So there's a whole little realm of questions, issues, puzzles, concerns that shows up in real life, for example, when you pack the trunk for a ski vacation and you got all these objects of varying sizes and shape and you got this space to put them in and they don't fit. Well what if we take the skis out and turn the lug it that oh then it goes under this and then that can slip in, you know, that kind of satisfaction? So that's what mathematicians call a packing puzzle. And since mathematics is kind of our idealized distillation of the pure underlying patterns of nature, music, art, you know, reality, but then abstracted just as painting starts as representation of reality and then starts leaving. We could imagine a kind of an idealized simplified purified packing puzzle. And I wanted to give you examples that could be be talked about for an audio only audience. Okay, great. And every mathematician when they hear that goes ouch I can't even draw it's going to be that's going to put a ceiling on how deep this can get we have YouTube too though so people will be able to I'm going to draw when I when I can but also I hope that I can get across this anyway. So one day my son and I, he was I don't know eight, seven, we're sitting on the floor and there's toys around and if you have a mathematical worldview, a kid's toy is a source of all sorts of things. So in this case it happened to be a box of dominoes and they were on the ground and for the purposes of this example we're not talking about the dots on the dominoes or anything like that, just the shape. Basically a one by two rectangle. Okay. And we had a bunch of these and they're fun they're you know made of glass or plastic or ivory or whatever and you're moving them around they feel all slippery and they're kind of cool. These were very shiny slippery ones. And we started thinking about them as bricks. And you can start building things with them or whatever. And questions start emerging from that. So here's a question that emerges pretty naturally from just playing around with dominoes and see what you think of it. Suppose I have a square grid so I'm I'm just gonna make a very easy little four by four square grid like that. And your dominoes are sized so that they occupy or cover exactly two s of those grid squares. So you could for example make this design out of dominoes. You could you could pack the four by four grid beautifully and perfectly with dominoes without much difficulty. You see that? Yep. In fact you could do it in lots of different ways. Yes and who poof there a math problem how many ways? Whoa that's an interesting and deep question. You could try to list all however many thousand ways I don't want to do that. So the prop puzzle there would be how many ways are there to tile a four by four grid with dominoes without having to list them all? Can we figure that out? I don't know. That's on the side. But the puzzle that I wanted to talk about is this it's clear that you could tile this with dominoes for example make the dominoes vertical you could make them horizontal. It's it's almost effortless. Suppose I removed one of the squares. So now it's a four by four grid with a corner removed the rest of the shape is like a square with a little notch in it. Can you cover that with dominoes?

SPEAKER_04 34:58

I don't think so.

SPEAKER_02 34:59

Yeah it turns out if you try to cover this with dominoes you will always fail. And we could run through you know 738 whatever possible ways of arranging dominoes and see that they all fail. But that's gross. No one wants to do that. There's a much simpler maybe maybe it's already occurred to you maybe it's occurred to listeners who have even audio only why there's no way to do this. And it's because there's an odd number of squares now.

SPEAKER_04 35:24

Right. You're taking them up two at a time and now you're yeah. So yeah.

SPEAKER_02 35:28

So that's uh that's a small little elegance there is oh it's an even and odd thing. All right well let's remove the other opposite corner. So now uh instead of a four by four grid with 16 squares we've taken two away so we've got 14 squares. So we're back to even. So good. So there's no obstruction from the even odd point of view of why we shouldn't be able to cover this with dominoes. So let me uh switch to a red pen. Suppose I try well let me put a domino here. So I'm using that line to indicate that those two squares are tied together as a domino. And obviously if I can overlap the dominoes or leave gaps then I can do whatever I want. So the the question is can I beautifully and harmoniously design some sort of way to tile this board and you're shaking your head which means you're seeing that it's awkward. Yeah I don't see it anywhere I'm not sure but maybe that's because I started out in a lame way.

SPEAKER_04 36:30

Yeah I was trying it in my head before you started drawing I Right these are fun puzzles.

Domino Tiling And Elegant Explanations

SPEAKER_02 36:34

You're engaged already right because math is about our curiosity it's it's almost the distilled essence of human curiosity. The human brain is a pattern recognition machine. This is what we do. We're biochemical mathematicians whether we know it or not. Okay so it's less clear whether this board can be covered and if the board were larger if we went to say an eight by eight board with the opposite corners removed there's going to be millions maybe literally millions of ways to orient the dominoes try, fail, maybe succeed. We have a mystery is what we have two mysteries really the mystery is is it possible? Okay? If it's possible we might stumble upon it fantastic great but let's suppose it turns out to be impossible. Then what's the mystery? Mystery is every five year old's favorite word. Why? Why? Why why why why should it be impossible? We saw why removing one corner was impossible. We had a good rationale dominoes cover two squares at a time there was an odd number no way it's going to work but here there's an even number there's no sort of moral reason why this shouldn't work. And yet it's not working. So we want an explanation we can say I tried every single one the fact is it doesn't work well okay but that doesn't tell me anything about an eight by eight board you want to try all those? Okay so here's a piece of mathematics suppose we were to audaciously take our board which maybe I want to figure out how to do this I'm gonna try to maybe redraw our board a little bit here probably in the least efficient possible way. Here's the board we want to cover right yeah the four by four corners grid with the corners removed. And you see we could ask the same question really about any size grid. It could be six by six could be whatever you could even say why does it have to be a square why can't it be a six by eight yeah right why can't it be three D we're using three D dominoes. Sure that's what mathematicians do we take things we generalize etc but suppose that I took this board and I actually thought of it as being part of a checkerboard and I actually checkerboard color it let me use red and white so I've got a red and white checkerboard grid and I just ran out of ink that's just great.

SPEAKER_04 39:12

Okay now were that the original four by four board right it would be 16 squares but checkerboard gridded so there'd be eight red and eight white yeah but what happened is I removed two whites so this board is actually eight reds six white and you always need the even number of reds and an even number of whites.

SPEAKER_02 39:34

Well what happens when you put a domino down on a board like this it covers one of each. Right. It doesn't just cover two squares it covers one of each color so if your board isn't color balanced there's no hope. So that's a gorgeous explanation that works for a 20 billion by 20 billion size board with the corners removed. I don't even need to try one single example to know it's impossible because this checkerboarding idea and here's the key phrase allowed me to see mathematics is the art of seeing the art of explaining why in the simplest most elegant most revelatory possible way I'm not saying this is a big important landmark in the history of mathematics it isn't. It's a small little problem in combinatorial geometry with a very pretty elegant little solution that by the way would continue to work not only for every size board but would work in three dimensions if we were making cubic grids and having dominoes made of two cubes, they'd still cover one of each color. We could still talk about color balance.

Seeing Area Instead Of Memorizing

SPEAKER_04 40:53

Well this is so great. I mean it's a beautiful example I want to kind of bring this one back because you did this thing in the book where you drew this dotted line right down the middle of the triangle. And now you can see easily that the the triangle is half of this one and half of this one like this this rectangle split in half and this rectangle split in half just drawing that dashed line allows me to see and now all of a sudden I'm at area of a triangle equals one half base times height. And you use this example in the book of how we've told everyone area is one half base times height and they go around memorizing it and then it's like this useless annoying thing that I'm just like I'm troubled by my whole life without any curiosity never the invitation to look it up examples better than you even know because when you say area is blah blah blah based on side thing no area is the amount of space inside it's not a number it's not a formula it's an amount of space and we can ask interesting questions.

SPEAKER_02 41:58

A very deep one would be how much of a cube does a sphere take up if you have a basketball in a box is it half? More than half? Less than half? Can humans know? All right so all of these beautiful classical problems that had you know Plato and Archimedes you know walking around musing and thinking hard these are deep beautiful simple elegant incredibly difficult problems to solve. The practical problem of how much does a basketball take up of a box is easy. Fill it with water dump it out measure it you because you all you care about is an approximate rough thing because you've got a rough box and a rough ball. But if you have a perfect imaginary cube with a perfect imaginary sphere in it, how much space does it take up is a perfect number. Is it half turns out not it's pretty close but not can we know anything about it? And this is what gets mathematicians excited and thinking another example this is really lowly but I think will be understandable to almost anyone and also goes to the heart of why some of the the this cultural stuff is so problematic five times three is three times five. Who says why five baskets with three eggs each is that the same scenario as three baskets with five eggs each not to me why should those be have anything to do with each other at all because we're trained that five times three is three times five is fifteen I memorized it well maybe there's something more to be said there because five copies of three and three copies of five are two completely different scenarios. So that's an example of a setting that requires an explanation. We need a piece of mathematics here. I can give it to you and I hope that you We'll find it charming because um this is might be the first piece of mathematics that was ever done by any human being was the explanation of why three groups of five and five groups of three are the same. And you see this as three rows of five? Yes. You see it as five columns of three? Yes. Done. We did it. You can turn your head. Right. And that's why three fives is five threes. But that needed to be made. Just like that needed to be made. The picture helps you see. We need to lift the veil somehow. And the way we do that is with rational argument. So it's the art of reason.

Kindling Desire Instead Of Pouring Facts

SPEAKER_04 44:47

It's beautiful. It's beautiful. So good. Where where were you when I was a child? Is my question. I needed you. Where was I when I was a child? Where were you when you were a child? Um, you know, one of the things we talk about, and this this theme is coming up over and over again. You probably noticed the podcast we called the Kindled Podcast. This is a reference to Plutarch. I'm sure you've heard this quote before. There's kind of two paradigms of thinking about human learning, and one of them is about lighting a fire, and one of them is about filling a cup with water. And I think so much of the structure, everything we've been talking about is is pouring water. The area of a triangle is calculated with one and a half base times height, and five times three is equal to three times five, and that's 15. It's like we're hitting them with all of this water. It's like here, you gotta hit this. What would you I I think both of us would would be in agreement that pouring water on a fire is a bad idea. We gotta stop it. We we gotta let the fire spark and kindle.

SPEAKER_02 45:42

Do you happen to have a copy of a mathematician's lament?

SPEAKER_04 45:44

I do.

SPEAKER_02 45:45

Easy grasp.

SPEAKER_04 45:46

Yeah, it's right here.

SPEAKER_02 45:47

Can you open it to the very per first page with the quote from Antoine de Saint-Ex?

SPEAKER_04 45:54

Yeah, let's see. I'm I'm on a digital copy, so I don't know if I have it.

SPEAKER_02 45:58

Okay.

SPEAKER_04 45:59

Well, it would be like it starts right into a musician wakes from a terrible nightmare. Oh, okay.

SPEAKER_02 46:05

So you're not looking at the book.

SPEAKER_04 46:06

No, no, no.

SPEAKER_02 46:06

It's a Oh, you're looking at the article. Yeah. Okay, that's fine. Well, I start the book with a quote from the author of The Little Prince. And I'll have to just kind of wing it. But it's like if you want to build a ship, don't gather people and s and uh start assigning tasks. Teach them to long for the immense beauty of the sea. That's how you get a ship built. Yeah.

SPEAKER_03 46:35

Yeah.

SPEAKER_02 46:36

And that's my version of the Blue Tar thing.

SPEAKER_03 46:39

It's beautiful.

SPEAKER_02 46:42

Be a person who loves what they love and can express it. I used to make this joke about I've got a two-point, two-part program to be an awesome math teacher. Step one, spend 40 years, six hours a day, every day, doing mathematics. Step two, tell kids about your experience. So I know that's a little mean, but that's basically where I'm coming from.

SPEAKER_04 47:12

Right.

SPEAKER_02 47:12

Is I'm just talking about myself, my art, the things I love. So it's effortless. I've never prepared a class in my life. No. I don't want to. I wouldn't want to have a script. I get that for a lot of teachers, it's a shield. It protects you. The textbook is part of your shield, the lesson plan, the notes, so that you get to write on the blackboard exactly what was in the textbook and they get to write it in their notes and it's all the same and what what happened, nothing. I get that that protects you from your vulnerability, but you're going to have so much more fun if you're just honest and real and be a human and be vulnerable and not know the answer to their question. And then you get to go see, try to find out that answer.

Protecting Your Mind And Pursuing Happiness

SPEAKER_04 47:54

Yeah. And you're going to light fires in the process. Paul, this has been fascinating. I appreciate you just shedding light and making me more excited about math. Will you just give me, I, as kind of one final, I don't know, a perspective. And then I will invite you. We invite all of our guests to just share somebody in your life that's that's kindled the love of learning for you. But before you do that, will you just kind of paint a picture for me of what a world might be like if fire was burning or if everyone longed for the sea, if math wasn't done in the way it's done, where everyone grows up hating it and afraid of it and you know not really knowing even what it is at all. And instead, if everyone had not not to like put you on a pedestal, but if everyone had been in a class like yours where, you know, it is about the curiosity, the play, the fun, the questions, the patterns, all of that. How would the world be different in your mind? I mean, I I know that's a tricky question to answer.

SPEAKER_02 48:46

It is a tricky question. And I and I know that, you know, my my friends and students who who might be listening are bracing for a huge anti-capitalist Marxist diatribe and all that. I I'll forego that. Um, the trouble with the world is there's a lot of troubles with the world, as you know. And it's not just that people aren't doing math, they're also not writing poetry, they're not also not painting paintings, they're also not spending enough time in the park and outside and all that, because everyone's, you know, work it hard, make a living, pay the rent, blah, blah, blah, blah, blah, blah, blah. And, you know, I would like people to think harder about whether that's actually getting them anywhere in the happiness department. You know, the U.S. Constitution is a very radical document in some ways. And one big way is that the pursuit of happiness is mentioned explicitly as the goal. Whereas in 1968, if you're in Washington Square Park and you're talking about pursuing happiness, the police beat you up.

SPEAKER_03 49:51

Yeah.

SPEAKER_02 49:52

So we got to pursue happiness. And for me, mathematics is an enormous contributor to that. It's free, it's fun, it's in your head, you carry it around with you. I mean, I I'll say this. I think that if more people were doing mathematics as a species, we would be much happier and way smarter. Doing mathematics, just like being a musician opens your ears in ways that you're hearing things that people don't hear, doing mathematics opens your mental faculties in a way nothing else comes close to doing. Your ability to follow reasoned arguments, to construct them yourself, to reject lines of argument because they're too ugly, to develop your own taste and sophistication and technical skill if you want that. You know, I think we'd be way better off if a lot more people knew what math really was, knew that some of the things that they do enjoy in life, various kinds of puzzles. You know, in some ways, more people are doing math now than ever, because people are playing games on their phone. And some of those are word games with some deduction, but some of them are very purely mathematical. Very beautiful little gems of math, like the Rubik's Cube or Tetris, where you're doing something very satisfying that has to do with you going, no, if I move that one over there, there's not going to be room for that one. You're doing math there when you're doing that. I'm sorry that school ruined any pleasure that you might have in thinking about lines and circles and numbers and those puzzles and games, but you shouldn't let it. I recommend for all humans take a really defensive posture about your mind. Protect it. Protect it from church, protect it from school, protect it from the government. You're gonna have your life and you're gonna be on your deathbed, and you're gonna have to say, What did I do? What did I allow myself to be subjected to? God, I wish I had been a little more assertive of my autonomy.

The Seven-Year-Old Who Taught Him

SPEAKER_04 52:17

Yeah, you you want to use this uh this brain that you've been given. Well, Paul, this has been fascinating. Will you please take a minute and just share, if if you're willing, shout somebody out from your life that's helped kindle a love of learning for you? Because whatever they did, it worked, and here you are as a lifelong learner yourself, sharing it with others.

SPEAKER_02 52:35

I'm gonna pick a funny example because it's not a teacher of mine, it's a student of mine. At some point, I don't even know if I know what year it was, 2004, 5, 6, I don't know. Um We had a student at the school who was seven years old and was exhibiting signs of mathematical precocity that were extraordinarily unusual. So my friend Richard was teaching this second grade class, and he was doing some things with squares and chopping squares diagonally and making other little squares and things like that. Just, you know, a fun little exploration of shape. And this little boy started to he said, It's not gonna work. And Richard was like, What's not gonna work? And he started to understand what the kid was upset about. He had discovered the irrationality of the square rooted too. Whoa! And that was discovered because people were playing with mosaic tiles and the Pythagorean theorem and things like that to come out of you moving triangles and squares around and making designs. But it's very unusual for a person to see that at that age. And so what ended up happening is he was given to me for a one-on-one math relationship from the age of seven to the age of eighteen. So I got to have what Jean-Jacques Rousseau writes about in Emile. He was imagining that a friend of his gave him a child to educate and how he would do that. And that's a really, really interesting book to read, Emile, by the way, because it's a great counterpoint to the sit in a desk and look at a blackboard. It's uh be in the world, do life and learn everything from it. Fascinating. In any case, my experience with Nicholas from age seven to age eighteen was the most intense, mind-blowing teaching experience of my entire life. Because you think you understand what you understand, but when a small child pokes in the right place, you realize, oh dear, I've got some thinking to do about this. And so that was a miracle in my life.

Books Plug And Prenda Closing

SPEAKER_04 55:02

Beautiful. I hope in his too, but uh we had a blast. It sounds like it's a it's another book. I don't know if you're needing another thing to write about, but Paul Lockhart, this has been fascinating. I know our listeners are gonna be thrilled. I hope you're sitting there just dazed and confused a little bit and wondering about all of your life choices, you know, as you think about your own perspective on math and what you even think meth is. I hope you've had cause to pause a little bit and reassess as you listen to Paul here today. We invite everybody to continue to just ask questions, make it fun, and by all means get out of the way of kids when they're doing that, help them ask those questions and be supportive of that natural journey. I have to just isn't still.

SPEAKER_02 55:43

I promised my publisher, I have to quickly get in a quick plug a mathematician's lament, arithmetic, measurement, the mending of broken bones. Those are my books.

SPEAKER_04 55:52

Read Paul's books, you will be better for it. And thank you so much, Paul Eckhart, for being here.

SPEAKER_02 55:56

Thanks, Kelly. It was great fun.

SPEAKER_00 55:58

The Kindled Podcast is brought to you by Prenda. Prenda makes it easy to start and run an amazing microschool based on all the ideas we talk about here on the Kindled Podcast. Don't forget to follow us on social media at PrendaLearn. And if you'd like more information about starting a microschool, just go to Prenda.com. Thanks for listening, and remember to keep Kindling.