Jo Boaler: How to Learn Math | Lex Fridman Podcast #226

The following is a conversation with Joe. Bowler, a mathematics educator at. Stanford and co-founder of ukubed.org. that seeks to inspire young minds with. the beauty of mathematics. To support. this podcast, please check out our. sponsors in the description. This is the. Lex Freedman podcast and here is my. conversation with Joe Bowler. What to you is beautiful about. mathematics? I love a mathematics that some people.

don't even think of as mathematics which. is beautiful creative. mathematics where we look at maths in. different ways. We visualize it. We. think about different solutions to. problems. A lot of people think of maths. as you have one method and one answer. And what I love about maths is the. multiple different ways you can see. things. different methods, different. ways of seeing different in some cases. different solutions. So that is what is.

beautiful to me about mathematics that. you can see and solve it in many. different ways and also the sad part. that many people think that maths is. just one answer and one method. So to you the beautiful the beauty. emerges when you have a problem with a. solution and you start adding other. solutions simpler solutions. Mhm. Uh weirder solutions, more interesting, some of that are visual, some of that. are algebraic, geometry, all that kind of stuff.

Yeah. I mean, I I always say that you. can take any maths area and make it. visual. And we say to teachers, give us your. most dry, boring maths and we'll make it. a visual, interesting, creative problem. And turns out you can do that with any. area of maths. And I think we've given. PE it's been a great disservice to kids. and others that it's always been numbers. lots and lots of numbers. Numbers can be. great but you can think about maths in. other ways besides numbers.

Do you find that most people are better. visual learners or is this just. something that's complimentary? What's the kind of the full spectrum of. students in the way they like to explore. math would you say? I mean there's. definitely people who come into the. classes I do who are more interested in. visual thinking and like visual. approaches but it turns out what the. neuroscience is telling us is that when. we think about maths there are two. visual pathways in the brain and we.

should all be thinking about it. visually. Some approaches have been to. say well you're a visual learner so. we'll give you visuals and you're not a. visual learner. But actually if you. think you're not a visual learner, it's. probably more important that you have a. visual approach so you can develop that. part of your brain. So you were saying that there's some. kind of interconnected aspect to it. So. the visual connects with the non-visual. Yeah. So this is what the neuroscience. has shown us that when you work on a.

maths problem, there are five different. brain pathways and that the most high. achieving people in the world are people. who have more connections between these. pathways. So if you see a maths problem with. numbers, but you also see it visually, that will cause a connection to happen. in your brain between these pathways. And if you maybe write about it with. words, that would cause another. connection. Or maybe you build it with. something physical, that would cause a. different connection. And what we want. for kids is we call it a.

multi-dimensional experience of math. seeing it in different ways, experiencing it in different ways that. will cause that great connected brain. You know, there's these stories of. physicists doing the same. I find. physicists are often better at building. that part of their brain of uh using. visualization for intuition building. because you ultimately want to. understand. the like the deepest secret underneath. this problem and for that you have to. intuit it your way there. Yeah. And you you mentioned offline that um.

one of the ways you might approach a. problem is to try to tell a story about. it. And some of it is like legend, but I'm. sure it's not always is, you know, you. have Einstein. uh thinking about a train, you know, and the speed of light and you. know that kind of intuition is useful. You start to like imagine a physical. world like how does this idea manifest. itself in the physical world and then. start playing in your mind with that. physical world and think is this going. to be true? Is this going to be true?

Right. Right. Einstein is well known for. thinking visually. and people talk about how he really. didn't want to go anywhere with problems. without thinking about them visually. But the other thing you mentioned that. sparks something for me is thinking with. intuition. Like having intuition about. maths problems. That's another thing. that's often absent in maths class. The. idea that you might think about a. problem and use your intuition. Um but. so important. And when mathematicians.

are interviewed, they will very. frequently talk about the role of. intuition in solving problems but not. commonly acknowledged or brought into. education. Yeah. I mean that's what it is like if. if you task yourself well building an. intuition about a problem that's what. that's where you start to pull in like. um what is the pattern I'm seeing? In. order to understand the pattern, you. might want to then start utilizing.

visualization. But ultimately, that's all in service of. like. solving the puzzle like cracking it open. to get the simple explanation of what. why things are the way they are as. opposed to um. like you said having a particular. algorithm that you can then execute to. solve the problem. Yeah. But it's hard. It's hard. Like. reasoning is really hard. Yeah. It's it's hard. I mean, I love to. value what's hard in maths instead of. being afraid of it. We know that when.

you struggle, that's actually a really. good time for your brain. You want to be. struggling when you're thinking about. things. So, if it's hard to think. intuitively about something, that's. probably a really good time for your. brain. I I used to work with somebody. called Sebastian Thr who. it's a great sort of mathematician, you. might think of him, AI person. And I. remember in one interview I did with. him, he talked about how they'd built. robots, I think, for the Smithsonian and. how they were having this trouble with.

them picking up white noise. Mhm. [clears throat]. And he said they had to solve it. They. had to work out like what's going on and. how he intuitively. worked out what the problem was, but. then it took him three weeks to show it. mathematically. I thought that was. really interesting that how you can have. this intuition and know something works. It's kind of different from going. through that long mathematical process. of proving it. But so important. Yeah, it's I think probably our brains.

are evolved as like intuition machines. and the uh the the math of like showing. it like formally is probably an extra. thing that we're not designed for. You. see that with Fineman and the his. um I mean it just all of these. physicists definitely you see um. starting with intuition sometimes. starting with an experiment. and then the experiment inspires. intuition but you can think of an. experiment as a kind of visualization.

right. just like let's let's take whatever the. heck we're looking at and draw it. and and draw like uh the pattern as it. evolves as the thing grows for n= 1 for. n= 2, n= 3 and you start uh to play with. it and then in the modern day which I. loved uh doing is you know you can write. a program that then visualizes it for. you right. then you can start exploring it. programmatically. Mhm. And that that and then uh. you can do so interactively too. I I.

tend to not. uh like interactive because it way takes. way too much work because you have to. click and move and stuff. I love to. interact through writing programs but. that's my particular brain software. engineer. So like you can you can uh do. all these kinds of visualizations. Uh. and then there's the tools of. visualization like color uh all those. kinds of things. that you're you're absolutely right. They're actually not taught very much. like the art of visualization. not taught. And we love as well color.

coding. Like when you represent. something mathematically, you can show. color to show the growth. Yeah. And kind of code that. So if I have an. algebraic expression for a pattern, maybe I show the X with a certain color, but also write in that color. So you can. see the relationship. Very cool. And um. yeah, we particularly in our work with. elementary teachers, many of them come. to our workshops and they're literally. in tears when they see things making.

sense visually. because they've spent their whole lives. think not realizing you can really. understand things with these visuals. It's quite powerful. You say that u there's something about. there's something valuable to learning. when the thing that you're doing is. challenging, is difficult. So, a lot of. people say, you know, math is hard or. math is too hard or too hard for me. Do. you think math should be easy or should.

it be hard? I think it's great when things are. challenging, but there's something that. that's really key to being able to deal. with challenging maths, and that is. knowing that you can do it. And I think. the problem in education is a lot of. people have got this idea that you're. either born with a maths brain or you're. not. So when they start to struggle, they think, "Oh, I don't have that maths. brain." And then they will literally.

sort of switch off in their brain and. things will go downhill from that point. So struggle becomes a lot easier and. you're able to struggle if you don't. have that idea. But you know that you. you can do it. You have to go through. this struggle to get there. But you're. you're able to do that. And so we're. hampered in being able to struggle with. these ideas we've been given about what. we can do. Ask a difficult question. here. Yeah. So there's kind of um I don't know what.

the right term is, but some people are. um struggle with learning in different. ways like their brain is constructed in. different ways. And um how much should. as educators should we make room for. that? So how do you know the difference. between this is hard and I don't like. doing hard things versus my brain is. wired in a way where I need to learn in. very different ways. I can't learn it. this way. How do you find that line? How.

do you operate in that gray area? So. this is why being a teacher is so hard. and people really don't appreciate how. difficult teaching is when you're faced. with I don't know 30 students who think. in different ways and um but this is. also why I believe it's so important to. have this multi-dimensional approach to. maths. We've really offered it in one. [clears throat] way which is here's some. numbers and a method you follow me do. what I just did and then reproduce it.

And so there are some kids who like. doing that and they do well and a lot of. kids who don't like doing it and don't. do well. But when you open up maths and. you give you let kids experience it in. different ways maybe visually with. numbers with words what happens is kids. there are many more kids who can access. it. So those different brain wirings you're. talking about where some people are just. more able to do something in a. particular way. That's why we want to.

that's one of the reasons we want to. open it up so that there are different. ways of accessing it and then that's not. really a problem. So I uh grew up in the Soviet Union and. uh fell in love with math early. I was. forced into math early and fell in love. through force. That's good. Well, [laughter] good that. you fell in love about the force. Well, but there uh something we we talked. about a little bit is there's such a.

value for excellence. Uh it's. competitive and it's also everybody kind. of looks up the the definition. of success is being uh in a particular. class is you know being really good at. it. Mhm. Um and like it's not improving, it's. like being really good. I mean we are. much more like that with sports for. example. We're not. It's like it's. understood, you know, you're going to. star on the basketball team if uh you're.

going to start on the basketball team if. you're going to be better than the other. guys, the other girls on the team. Mhm. Uh so. that coupled with the belief, this could. be partially a communist belief, I don't. know, but the belief that everybody is. capable of being great, but if you're. not great, that's your fault and you. need to work harder. And I remember I. had a sense that um probably delusional, but I could win a Nobel Prize. I don't. even know what that entails.

Um, but I thought um like uh my dad. early on told me just offhand and it. always stuck with me that if you if you. can figure out how to build a time. machine, how to travel back in time, it. will probably give you a Nobel Prize. And I remember early in my life thinking. I'm going to invent a time machine. And. like like the tools of mathematics were. in service of that dream of winning the. Nobel Prize in it's silly. I didn't.

really think in those concrete terms, but I just thought I could be great. That feeling. and then then when you struggle, the. belief that you could be great is like. struggle is good that. right pushes you on. Yeah. And so the other thing about the Soviet. system that might love to hear your. comments about is just the sheer like. hours of math. like the number of. courses you're talking about. a lot of geometry a lot more ge I think. in the American system you take maybe.

one year of geometry. in high school yeah. in in high school first of all geometry. is beautiful it's visual and then you. get to reason through proofs and stuff. like that in in Russia I remember just. being nailed over and over with it was. just non-stop. and then of of course there's different. perspectives on calculus and just the. whole the sense was. that math is like like fundamental to. the development of the human mind. So. math but also science and literature by.

the way was also hit very hard. Like we. read a lot of serious adult stuff. America does that a little bit too. They. they challenge young adults with good. literature but they don't challenge. adults very much with math. Math. So those two things um valuing. excellence and and just a lot of math in. the curriculum. Do you think do you. think do you find that interesting. because it seems to have been. successful. Yeah, I think that's very interesting.

and there is a lot of success people. coming through the Soviet system. I. think something that's very different to. the US and other countries in the world. is that idea that excellence is. important and you can get there if you. work hard. In the US, there's an idea. that excellence is important, but then. kids are given the idea in many ways. that you can either do it or you're one. of the people who can't. So many. students in the school system think. they're one of the kids who can't. So.

there's no point in trying hard because. you're never going to get there. So if. you can switch that idea, it would be. huge. And it seems from what you've said. that in the U in the uh Soviet Union. that idea is really different. Now the. downside of that idea that anybody can. get there if you work hard is. that thought that if you're not getting. there it's your fault. And I I would add. something into that. I would say that. anybody can get there,

but they they need to work hard and they. also need good teaching. because there are some people who really. can't get there because they're not. given access to that good teaching. So, but that would be huge that change as to. doing lots of maths. If um if maths was. interesting and open and creative and. multi-dimensional, I would be all for. it. We we actually run summer camps at. Stamford where we invite kids in and we. give them this maths that I love. And.

the in our camp classrooms, they were. three hours long. And when we were. planning, the teachers were like, "Three. hours? Are we going to be able to keep. the kids excited for 3 hours?" Turned. out they didn't want to go to break or. lunch. They'd be so into these. mathematical patterns. We couldn't stop them. It was amazing. So yeah, if maths was more like that, then I think having more of it would be. a really good thing. So what uh what age are you talking.

about? Is there um could you comment on. what age is like the most important when. people quit math or give up on. themselves or on math in general and uh. perhaps that age or something earlier is. really important moment for them to. discover to be inspired to discover the. magic of math. I think a lot of kids. start to give up on themselves and maths. around. from about fifth grade and then those. middle school years are really.

important. Wow. And fifth grade can be pivotal for kids. just because they're allowed to explore. and think in good ways in the early. grades of elementary school. But fifth. grade teachers are often like, "Okay, we're going to prepare you now for. middle school and we're going to give. you grades and lots of tests." And. that's when kids start to feel really. badly about themselves. And so middle. school years, we our camps are middle. school students. We think of those years. as really pivotal. Many kids in in those.

years are deciding, yes, I'm going to. keep going with STEM subjects or no, I'm. not that this isn't for me. So I mean. all years are important and in all years. you can kind of switch kids and get them. on a different pathway but I think those. middle school years are really. important. So what's the role of the. teacher in this? So one is the. explanation of the subject but do you. think teachers should almost do like. one-on-one. you know little Johnny I believe in you.

kind of thing like that that energy of. like. turns out it's really important. There's. um a study that was done. It was. actually done in high school English. classrooms where all kids wrote an essay. for their teacher and this was done as. an experiment. Half of the kids got. feedback from their teacher, diagnostic. feedback, which is great. But for half. of the kids, it said an extra sentence. at the bottom that the researchers had. put on. Mh. And the kids who read that extra. sentence did significantly better in.

English a whole year later. The only. change was this one sentence. What did the sentence say? So what did [laughter] the sentence say? The sentence said, "I'm giving you this. feedback because I believe in you." And. the kids who read that did better a year. later. Yeah. So when I share this with teachers, I. say, you know, I'm not suggesting you. put on the bottom of all kids work. I'm. giving this feedback cuz I believe in. you. One of the teachers said to me, we. don't put it on a stamp. I said, "No,

don't put it on a stamp. It's um but. your words are really important and kids. are sitting in classrooms all the time. thinking, what does my teacher think of. me? Does my teacher think I can do. this?" Um so it turns out it is really. important to be saying to kids, I know. you can do this and those messages are. not given enough by teachers. and really believe it. And believe it. Yeah. It's like.

you can't just say it. You have to. believe it. I I I sometimes cuz like. it's it's such a funny dance cuz I'm. such a perfectionist. I'm I'm extremely. self-critical and I have when I have. students come up to me and it's clear to. me that they're. not even close to good. And it's. tempting for me to be like uh to sort of. give up on them mentally. But the. reality is like if you look at many. great people throughout history, they. sucked at some point.

Yeah. Exactly. And and some of the greatest took. nonlinear paths to where they sucked for. long into into later life. And so always kind of believing that. this person uh can be great. Exactly. That. you have to communicate that plus the. fact that they have to work hard. That's it. Yeah. Yeah. And you're right. Silicon Valley where I live is filled. with people who are dropouts at school. or who had special needs who didn't. succeed. Um it's very interesting that.

have gone on to do amazing work in. creative ways. I mean I do think our. school system is set up to um value. good memorizers who can reproduce what a. teacher is showing them and push away. those creative deep thinkers. often. slower thinkers. They think slowly and. deeply and they often get the idea early. on that they can't be good at maths or. other subjects. So um yeah, I think many of those.

subject people are the ones who go on. and do amazing things. So there's a guy. named Eric Weinstein. I know many. mathematicians like this, but he he. talks a lot about not having a about. having a non-standard way of learning. M. I mean a lot of great mathematicians, a. lot of great physicists are like that. And he felt like he became quickly he he. got his PhD at Harvard became quickly an. outcast of the system like the the.

education especially early education. system didn't help him. Mhm. Um is there ways for an education system. to support people like that? Is it this. kind of multi-dimensional learning that. you're. me? Absolutely. Absolutely. I mean, I. think our education system still uses an. approach that was in classrooms hundreds. of years ago. The textbooks have a lot. to answer for in producing this very. uninspiring mathematics. Um, but yeah, if you open up the subject and have.

people see and solve it in different. ways and value those different ways. Somebody I appreciated a lot is a. mathematician called Mariam Mizakani. I. don't know if you've heard of her. She. won the Fields Medal. She was from Iran. Mhm. Uh first woman in the world to win the. Fields Medal in mathematics. She's she. died when she was 40. She was at. Stanford. But her work was entirely. visual. And she talked about how her. daughter thought she was an artist. because she was always visualizing. And.

I attended she asked me to chair the PhD. defense for one of her students. And I. went to the defense in the math. department. And it was so interesting. because this young woman spent like 2. hours sharing her work. All of it was. visual. In fact, I don't think I saw any. numbers at all. It's awesome. And I remember that day thinking, "Wow, I could have brought a like 13-year-old. into this PhD defense. They would not. recognize this as maths.". Mhm. But when Mariam Ms. Carney won the.

Fields Medal, all these other. mathematicians were saying that her work. had connected all these previously. unconnected areas of maths. And so, but. when she was she also shared that when. she was in school, when she was about. 13, she was told that she couldn't do. maths. She was told that by her teacher. This is Iran. She grew up. in Iran. Yeah. So, I love that. You. know, to be told you can't be good at. maths and then go on and win the Fields.

Medal is cool. I've been told by a lot of people in my. life that I can't do something. I'm very. definitely non-standard. Um, but all it. takes is that's that's why people talk. about like the one teacher that changed. everything for them. All it takes is one. teacher. That's right. That's that's the power of that. Mhm. that. [laughter]. So that that's like that should be. inspiring to teachers like. I think it is. you as a single person given the. education system given the incentives.

you have the power to truly change lives. and like 20 years from now I feel as. medalists will walk up to you and and. say thank you. you did that for me. Yeah, absolutely. And I share that that with teachers that. even in this broken system of what they. have to do for districts and textbooks, a single teacher can change kids maths. relationship or other subjects and uh. forever. What's the role of the parents in this. picture? Let's go to another difficult.

subject. Yeah, that is a difficult subject. Um, one study found that. um, the amount of maths anxiety parents. had predicted their child's achievement. in school. Um, but. only if they helped with homework. So, [laughter]. oh, that's so funny. Yeah, but there are some interesting. implications for this. I mean, you can.

see how it works. So, if you have maths. anxiety and you're helping your kids. with homework, you're probably. communicating things like, "I was. terrible at this at school and and. that's how it gets passed on to kids.". So, one implication is if you have a. really bad relationship with maths, you. hate maths, you have math anxiety, just. don't don't do maths with your kids. Um, but we have a on our website we have a. little sheet for parents of ways to.

interact around maths with your kids. And. that's uh youubed.org. That's ukubed.org. Yes. So, one of the. things I say to parents when I give. parent presentations is even if you hate. maths, you need to just fake it with. your kids. you should be always. endlessly optimistic and happy about. doing maths. And um. I'm always curious about this with this. So I you know I hope to have kids one. day. I don't have kids currently.

Um are parents [clears throat] okay with. like sucking at math and then trying to. get their kid to be better than them. essentially? Like is that difficult. thing for a lot of parents? It is. difficult. to have like it's almost like an ego. thing like I never got. good at this and I probably should have. and. yeah I mean to me that's you want to. celebrate that but I know a lot of. people struggle with that like coaches. in sports to to make an athlete become. better than them it can be hard on the.

ego. Yeah. So is that do you experience the same. with parents. too? I think I mean I haven't. experienced parents worrying that their. kids will be better than them. I have. experienced I have experienced parents. just having a really bad relationship. with maths and you know not wanting to. help not knowing how to help saying. things like another study showed that. when mothers say to their daughters I. was bad at maths in school their.

daughter's achievement goes down. So, we. know that kids pick up on these messages. and um which is why I say you should. fake it. But also, I know that lots of. people have just had a really bad. relationship with maths. Even successful. people like the undergrads I teach at. Stanford have pretty much always done. well in maths. But they come to Stanford thinking maths. is a set of methods to memorize. And so, so do many parents believe that.

There's one method that you memorize and. then you reproduce it. So until people. have really had an experience of what I. think of as the other maths where until. they've really seen that it's a really. different subject, um. it's hard for them to be able to shift. their kids to see it differently. Is. there for a teacher if we're to like. systematize it, is there something. teachers can do to do this more.

effectively? So you you mentioned the. textbook. So So what what are the. additional things you can add on top of. this whole old school traditional way of. teaching. that can improve the the process? So I do think there's a way of teaching. maths that changes everything for kids. uh and teachers. So I'm one of five. writers of a new framework for the state. of California, a new maths framework. It's coming out next year and we are.

recommending through this maths. framework that people teach in this way. It's called teaching to big ideas. Mhm. So um at the moment people have. standards that have been written and. then textbooks have taken these. standards and made not very good. questions. Mhm. And if you look at the standards, like I. have some written down here, just. reading the standards, it makes it math. seem really boring and uninspiring. What what what are the kind of Can you. give a few examples? What.

So this is an an interesting example. In. third grade, there are three different. standards about unit squares. [laughter and snorts]. Okay. And so this is one of them. A square. with side length one unit called a unit. square is said to have one square unit. of area and can be used to measure area. and that's something you're expected to. learn. that is something that so that's a. standard the textbook authors say I'm. going to make a question about that and. they translate the standards into narrow.

questions. and then you measure success by your. ability to deliver on on these. standards. So the standards themselves. uh I I think of maths and many people. think of maths in this way as a subject. of like a few big ideas and really. important connections between them. Um. so like an you could think of it as like. a network map of ideas and connections. And what standards do is they take that. beautiful map and they chop it up like. this into lots of little pieces and they. deliver the pieces to schools. And so.

teachers don't see the connections. between ideas nor do the kids. So. anyway, this is a bit of a long way of. saying that what we've done in this new. initiative is we have set out maths as a. set of big ideas. and connections between them. So this is. um grade three. Mhm. So [clears throat] instead of there. being 60 standards, we've said, well, you can pull these different standards. together in with each other and um also.

value the ways these are connected. And. by the way, for people who are just. listening, we're looking at a small. number of uh like big concepts within. mathematics, square tiles, measuring, fraction, shape, and time, and then how. they're interconnected. Mhm. And so the goal is for this is for grade. three, for example. Yeah. And so we've set out for the state of. California, the whole of mathematics K.

10 as a set of big ideas and. connections. So we know that teachers it. works really well if they say okay so a. big idea in my grade is measuring. Um and instead of reading five. procedural statements that involve. measuring they think okay measuring is a. big idea. What rich deep activity can I. use that teaches measuring to kids? And. as kids work on these deep rich. activities, maybe over a few days, turns.

out a lot of maths comes into it. So we're recommending that let's not. teach maths according to all these. multiple multiple statements and lots. and lots of short questions. Instead, let's teach maths by thinking about what. are the big ideas and what are really. rich deep activities that teach those. big ideas. So that's the like how you. teach it and maximize learning. What. about like from a school district.

perspective. like measuring how well you're doing, you know, grades and tests and stuff. like that. Do you throw those out or is. there is it possible? I'm not a fan of grades and tests um. myself. I think grades are fine if they're used. at the end of a course. So at the end of. my maths course, I might get a grade. because a grade is meant to be a. summitative measure. It kind of. describes your summit of achievement. But the problem we have in maths. classrooms across the U the US is people.

use grades all the time, every week or. every day even. My own kids when they. went through high school, technology has. not helped with this. When they went. through high school, they knew they were. being graded for everything they did. Everything. And not only were they being. graded for everything, but they could. see it in the grade book online and it. would alter every class they went into. So this is the ultimate what I think of. as a performance culture. You're there. to perform, somebody's measuring you,

you see your score. Um, so I think. that's not conducive for deep learning. And yes, have a grade at the end of the. year, but during the year you can assess. kids in much better ways. Like teachers. can a great way of assessing kids is to. give them a rubric that kind of outlines. what they're learning over the course of. a unit or a few weeks. So kids can. actually see the journey they're on like. this is what we're doing mathematically.

Sometimes they self assess on those. units and then teachers. um will show where what they can what. the kids can do with a rubric and also. write notes like you know in the next. few weeks you might like to learn to do. this or um so instead of kids just. thinking about I'm an A kid or a B kid. or I have this letter attached to me. they're actually seeing mathematically. what's important and they're involved in. the process of knowing where they are.

mathemat mathematically. at the end of the year, sure, they can. have a grade, but um during the year. they get these much more informative. measures. I I do think [sighs] this this might be. more for college, but maybe not. I some. of the best classes I've had is when I. got special like set aside. like the the professor clearly saw that. I was interested in some aspect of a. thing.

and then um I have a few in mind and one. in particular but he said that um he. kind of challenged me so this is outside. of grades and all that that kind of. stuff that. basically it's like reverse psychology I. don't I don't think this can be done and. so I gave everything to do that. particular thing. So this was happened. to be in an artificial intelligence. class. but I I think that like special. treatment of taking students. who are especially like excellent at a.

particular little aspect that you see. their eyes light up. I I often think. like maybe it's tempting for a teacher. to think you've already succeeded there, but they're actually signaling to you. that like you could really launch them. on their way. Yeah. Um and I don't know, that's too much to. expect from teachers, I think, to to to. pay attention to all of that cuz it's. really difficult. But I I just kind of. remember who are the biggest.

the most important people in in the. history of my life of education. And. it's those people that who really didn't. just like inspire me with their. awesomeness, which they did, but also. just they pushed me a little like they. gave me a little push. And that requires focusing on the quote. unquote excellent. uh students in the class. Yeah. I think what's important though is. teachers to have the perspective that. they don't know who's going to be. excellent at something before they give. out the activity.

Exactly. And in our camp classes that we. ran, um, sometimes students would finish. ahead of other students and we would say. to them, can you write a question that's. like this but different? Um, or and and over time we encourage them. to like extend things further. I. remember we were doing one activity. where kids were working out the borders. of a square and how big this border.

would be in different case sizes and one. of the boys came up at the end of the. class and said I've been thinking about. how you do this with a pentagon and I. said that's fantastic how do you how. what does it look like with a pentagon. go you know find out see if you can. discover so I didn't know he was going. to come up and say that and I didn't. have in my head like this is the kid who. could have this extension task, but you. can still do that as a teacher. Um, when.

kids get excited about something or. they're doing well in something, have. them extend it, go further. Mhm. It's great. And and then you also like this is like. teacher and coach, you could say it in different ways to. different students. Like for me, the. right thing to say is uh almost to say. uh I I don't think you could do this. This is too hard. like that's what I. need to hear. It's like no I you know. there's immediate push but with some. people if they're a little bit more. I mean it's all has to do with.

upbringing just how your genetics is. they might be much more that might break. them. Yeah that might break them. and so you have to be also sensitive to. that. I mean teaching is really. difficult. It [laughter]. is really difficult. for this very reason. It is. So, um, what is the best way to. teach math, to learn math at the at. those early few days when you just want. to capture them? I do something, actually, there's a. video of me doing this on our website. that I love when I first.

meet students. And this is what I do. I. show them a picture. This is the picture. I show them. And it's a picture of seven. dots like this. Mhm. And I show it for just a few seconds and. I say to them, "I'd like you to tell me. how many dots there are, but I don't. want you to count them. I want you to. group the dots." And I show it them and. then I I I take it away before they've. even had enough time to count them. And. then I ask them, "So, how did you see. it?" And I go around the room and.

amazingly enough, there's probably 18. different ways of seeing these seven. dots. Mhm. And so I asked people, "Tell me how. how you grouped it." And some people see. it as like an outside hole with a center. dot. Some people see like stripes. of lines. Some people see segments. And I collect them all and I put them on. the board. And at the end I say, "Look. at this. We are a class of 30 kids and. we saw these seven dots in 18 different. ways." There's actually a mathematical.

term for this. It's called groupetizing. [clears throat]. Groupetizing. Yeah. [laughter]. I like it. It's kind of cool. So, turns out though. that how well you groupize predicts. how well you do in maths. Is is it uh is it a raw talent or is it. just something that you can develop? I don't think it's I don't think you're. born groupetizing I think but some kids. have developed that um ability if you. like and you can learn it. you can. So,

this to me is part of how wrong we have. maths that we think to tell whether a. kid's good at maths, we're going to give. them a speed test on fact on multiples, but actually seeing how kids group dots. could be a more important assessment of. how [clears throat] well they're going. to do maths. Anyway, I diverge. What I. like to do though when I start off with. kids is show them, I'm going to give you. math problems. I'm going to value the. different ways you see them. And turns. out you can do this kind of problem. asking people how they group dots with.

young children or with graduate. students. And it's engaging for all of. them. [sighs and snorts]. Is um you talk about creativity a little. bit and flexibility in in your book. Limitless. What's the role of that? So. it sounds like there's a bit of that. kind of thing involved in. groupizing. Yeah. Yeah, I I love this term. So, what's what. would you say is the role of creativity. and flexibility in in the learning of.

math? I think what we know now is that what we. need for this 21st century world we live. in. is a flexible mind. It's school should not really be about. teaching kids particular methods, but. teaching them to approach problems with. flexibility. Being creative, thinking. creatively is really important. So. people don't think the words maths and. creativity come together, but I that's. what I love about maths is the creative.

different ways you can see it. And so. helping our kids, there's a book I like. a lot by physicists. You probably know this book called. Elastic. You might know it. Um and it's about how. we want elastic minds. Same kind of. thing. flexible creative minds and. schools do very little on developing. that kind of mind. They do a lot of. developing.

the kind of mind that a computer now. does for us. Memorization, memorization, doing procedures, a lot of things that we spend a lot of. time in school on. in the world when. kids leave school a computer will do. that and better than they will but that. creative flexible thinking we're kind of. at ground zero at computers being able. to engage in that thinking maybe we're a.

little above ground zero but um the. human brain is perfectly suited for that. creative flexible thinking that's what. humans are so great at so I would like. the balance to shift in schools maybe. you still need to do some procedural. kind of thinking But there should be a. lot more of that creative flexible. thinking. and uh what's the role of other humans. in this picture? So collaborative. learning so brainstorming together. So creativity as it emerges from the.

collective intelligence of multiple. humans. Yeah. Super important. And um we know. that also helps develop your brain that. social side of thinking. And I love. mathematics collaboration where people. build on each other's ideas and they. come up with amazing things. I actually. taught a hundred students calculus at. Stamford recently undergrads and we. taught them to collaborate. So these.

students came in Stanford and most of. them were against collaboration in. maths. This is before co in person. Yeah, it was just before co hit. It was. 2019. and um the summer. I'm sorry you said they're against uh. such. Yeah. So [laughter]. it's really interesting. So they had. only experienced maths individually in a. kind of competitive individual way and. if they had experienced it as group work. it had been a bad experience like maybe. they were the one who did it all and the. others didn't do much. So they were kind.

of against collaboration. They didn't. see any role for it in maths and we. taught them to collaborate. And it was hard work because as well as. the fact that they were kind of against. collaboration, they came in with a lot. of like social comparison thinking. So. I'm in this room with other Stamford. undergrads and they're better than me or. so when we set them to work on a maths. problem together. The first one was kind. of a disaster because they were all like. they're better than me, they're faster. than me. They came up with something I.

didn't come up with. Yeah. So we taught them to let go of. that thinking and to work well together. And one of the things we did, we decided. we wanted to do a pre and post test at. the end of this teaching. It was only. four weeks long, but we knew we didn't. want to give them like a time test of. individual work. So we gave them an. applied problem to do at the beginning. and we gave them to do in pairs together. and we gave each of them a different. colored pen and said work on this.

activity together and keep using that. pen. So then we had all these pieces of. student work and what we saw was they. just work on separate parts of the. paper. Uh there so this little like red. pen section and a green pen section. and they didn't do that well on it even. though it was a problem that middle or. high school kids could do but it was. like a problem solving kind of problem. and then we gave them the same one to do. at the end gave them the same colors and. it actually they had learned to. collaborate and not only were they.

collaborating the second time around but. that boosted their achievement and the. ones who collaborated did better on the. problem. Collaboration is important. having people and what was so eye. openening for these undergrads and they. talked about it in lovely ways was I. learned to value other people's thinking. on a problem and I learned to value that. other people saw it in different ways. and it was quite an experience for them. that they came out thinking you know I I.

can do maths with other people can see. it differently we can build on each. other's way ways thinking. I got a chance to I don't know if you. know who Daniel Conorman is got a chance. to interact with him. and like the first cuz he had a few but. one famous collaboration throughout his. life uh with Diverski and. just like you know he hasn't met me. before uh in person but just the number. of questions he was asking just the.

curiosity [snorts] so I think one of the. skills the collaboration itself is a. skill. And I remember my experience with him. was like, "Okay, I get why you're so. good at collaboration. because he was just extremely good at. listening. and genuine. curiosity about how the other person. thinks about the world, sees the world.". And then together he's he pulled me in. in that particular case. He doesn't know. in particular like that much about.

autonomous vehicles, but he kept like. asking all of these questions and then. like 10 minutes in we're together trying. to solve the problem of autonomous. driving. Mhm. And like and that I mean that's really. fulfilling. That's really enriching. But. it also in that moment made me realize. it's kind of a skill is you have to kind. of put your ego aside, put your view of the world aside and try. to learn how the other person sees it. And the other thing you have to put. aside is this social comparison. thinking.

Like if you are sitting there thinking, "Wow, that was an amazing idea. He's so. much better than I am." That's really. going to stop you taking on the value of. that idea. And so there's a lot of that going on. between these Stanford students when. they came. and. yeah, trying to help them let go of that. One. of the things I've discovered. just because being a little bit more in. the public eye, how rewarding it is to. celebrate others.

Yeah. And how much is going to actually pay. off in the long term. So this kind of silo thinking of like I. want to prove to a small set of people. around me that I'm really smart. and do so by basically not celebrating. how smart the other people are. That's. actually maybe shortterm it seems like a. good strategy but longterm it's not and. I think if you practice at the student. level and then at the career at every. single stage I think that's ultimately.

I I agree with you. I think that's a. really good way of thinking about it. You mentioned textbooks you and you. didn't say it uh you know um maybe. textbooks isn't the perfect way to teach. mathematics but I I love textbooks. There's they're like pretty pictures and. they smell nice and they open. I mean, I'm talking about like physical some of. my greatest experiences have been just. like oh like cuz they're really well. done. When we're talking about basic. like high school calculus,

biology, chemistry, those are like those. are incredible. It's like Wikipedia but with color and. and uh nice little. You must have seen some good textbooks. if they had pretty pictures and color. Yeah. Yeah, I mean I remember I guess it. was very very standard. like AP AP calculus, AP biology, AP. chemistry. I felt those are like some of. the happiest days of my life in terms of. learning was high school cuz it was it's. very easy honestly.

It felt hard at the time. But you're. basically doing a um whirlwind tour of. all of science. Yeah. Yeah. Without having to pick. You do like. literature, you do like Shakespeare, you. do calculus, biology, physics, u chemistry, what else? Anatomy, physiology, computer science without like nobody's. telling you what to do with your life. You're just doing all those things. That's a good thing. You're right. But I remember the textbooks weren't I.

mean, maybe I'm romanticizing the past, but I remember they they weren't they're. pretty good. Uh but so you think what. role do you think they play still and. like in this more modern digital age. what what's the best materials with. which to do these kinds of educations? Well I'm I'm intrigued that you had such. a good experience with textbooks. I mean. I can remember loving some textbooks I. had when I was learning and I love. books. I love to pick up books and look.

through them. But. a lot of maths textbooks are not good. experiences for kids. They um we have a. video on our website of the kids who. came to our camp and one of the students. says in maths you have to follow the. textbook. The textbook's kind of like. the Bible. You have to follow it. And. every day it's slightly different. Like. on Monday you do 2.3.2. 2 and on Tuesday you do 2.3.3 and on.

Wednesday and you never go off that. That's like every single day. and that's not inspiring for a lot of. the kids. So one of the things they loved about. our camp was just that there were no. books. Even though we gave them sheets. of paper instead, they still felt more. free because they weren't just like. trottting through exercises exercises. So um. like what what a textbook allows you is.

like you're. the the very thing you said they might. not like the 2.3. 2 point I it feels like you're making. progress and like it's little. celebrations cuz you do the problem and. it seems really hard and you don't know. how to do it and then you. try and try and then eventually succeed. And then you make that little step and. further progress and then you get to the. end of a chapter and you get to like. it's closure. You're like, "All right, I. got that figured out." And then you go.

on to the next chapter. I can see that. I mean, I think it could. be in a textbook. You can have a good. experience with a textbook. But it. what's really important is what is what. is in that textbook. What are you doing inside it? And. I mean, I grew up in England and in. England we learn maths. We don't have. this separation of algebra and geometry. And I don't think any other country. apart from the US has that. But I look. at kids in algebra classes where they're.

doing algebra for a year and I think I. would have been pretty bored doing that. [laughter]. Um, by the way, can we can we analyze your. upbringing real quick? Um, why do. British folks call mathematics maths? Why is it the plural? Is it is it. because of everything you're saying. where it's a bunch of subd disciplines? Yeah, I mean mathematics is sure is. supposed to be the uh the ma the.

different maths that you look at. whether you think of that as topics like. geometry and probability or or I think. of it as maths is just multi-dimensional. lots of ways but that's why it was. called mathematics and then it was. shortened to maths and then for some. reason it was just math in the US but to. me math has that more singular feel to. it and. there's an expression here which is do.

the math. which basically means do a calculation. that's what people mean by do the math. so I don't like that expression cuz you. know math could be anything doesn't have. to be calculation [clears throat] and so. yeah I like maths because it has more of. that broad feel to it. yeah I love that maths kind of. emphasizes the multi-dimensional like. the variety of different the different. subd disciplines different approaches. yeah. [laughter]. Okay. Uh what so but outside of the. textbook, what do you.

see like broadly being used? You. mentioned Sebastian Thran and MOO's. you know online education. Do you think. [clears throat] that's an effective set. can be. I mean online. having great teachers online obviously. extends those teachers to many more. people and that's a wonderful thing. Um, [clears throat] I have quite a few. online courses myself. I got the bug. working with Sebastian when he was had. released his first MOO and I thought,

hm, maybe I could do one in maths. education and I didn't know if anybody. would take it. Um, I remember releasing. it that first summer and it was a free. online class and 30,000 maths teachers. took it that first [laughter] summer and. they were all talking about it with each. other and sharing it and it was like. took off. In fact, it was that MOO that. c got me to create you cubed with Kathy. Williams who's the co-founder. Um. because people took the MOO and then. they said okay what now? Like I finished.

what what can I have next? And so that. was why we made our website. But um so. yeah, I think online education can be. great. I do think a lot of the muks. don't have great pedagogy. they're just. a talking ahead. and. you can actually engage people in more. active ways even in online learning. So I learned from the Udacity principle. when I was working at Udacity never to. talk more than like 5 minutes and but.

then and then to ask people to do. something. So that's the sort of pedigogy of the. online classes I have. There's a little. bit of presenting something and then. people do something and there's a little. bit more because I think if you have a. halfhour video you just you know switch. off and start doing other things. So the the way you did it is like 5 10. minute like bit of teaching. and then. with some visual stuff perhaps and then.

there's like a quiz almost. then you answer a question. Yeah. Yeah. Mhm. No, that that's that's yeah that's. really effective. You mentioned you. cubed. So what's the mission? What's the. goal? you you mentioned how it started, but what's uh. um yeah, [clears throat] where are you. at now and what do you what's your dream. with it or what are the kind of things. that people should go and check out on. there? Yeah, we started cubed I guess it was. about 5 years ago now and we've had over. 52 million visitors to the site. So I. very happy about that and our goal is to.

share good ideas for teaching with. teachers, students, parents in maths and. to help we have a sort of subgoal of. erasing maths anxiety that's important. to us but also to share maths as this. beautiful creative subject and um it's. been really great. uh we have lessons on. the site but one of the thing one of the. reasons I thought this was needed is. there's a lot of knowledge in the. academy about how to teach maths well.

loads and loads of research and journals. and lots of things written up but. teachers don't read it they they don't. have access to it they're often behind. pay walls they they're written in really. inaccessible ways so people wouldn't. want to read them or understand them so. this I see is a problem. You have this. whole industry of people finding out how. to teach well, not sharing it with the. people who are teaching. So, um that's.

why we made you cubed and instead of. just putting articles up saying here's. some things to read about how to teach. well, we translated what was coming from. research into things that teacher could. use. So lessons, there were videos to. show kids and um there were tips for. parents. There were all sorts of things. on the site and it's been amazing as we. had we took inspiration from the week of. code which. Mhm. got teachers to focus on coding for a. week and um we have this thing called.

the week of inspirational maths and we. say just try it for a week just just. give us one week and try it and see what. happens. And so it's been downloaded. millions of times. Teachers use it every. year. They start the school year with. it. And what they tell us is it was. amazing. The kids lights were on. They. were excited. They loved it. And then. the week finished and I opened my. textbooks and the lights went out and. they were not interested.

Yeah. But but getting that first. inspiration is still powerful. I mean, it is. I I wish I mean my what I would. love is if we could actually extend that. for the whole year. Mhm. We're a small team at Stamford and we're. trying to keep up with great things to. put on the site. Um we haven't the. capacity to produce these creative. visual math tasks for every year group. for every day but I would love to do. that. How difficult is it to do? I mean that's. to to come up.

[sighs and gasps]. with uh visual formulations of these uh. big important topics you need to think. about in a way that you know that uh. that that that you could teach. I mean it we can do it. We actually we. went from the week of inspirational. maths and we made K8 maths books with. exactly that big ideas, rich activities, visuals. We just finished the last one. We've been doing it for 5 years and it's. been exhausting and we just finished. So.

now there's a whole K8 set of books and. they're organized in that way. These are. the big ideas. Here are rich deep. activities. Mhm. Um they're not though what you can do. every day for a year that so some. teachers use them as a kind of. supplement to their boring textbook and. some people have said, "Okay, this is. the year. this book tells us what the. year is and then we'll supplement these. big activities with.

um so they're being used and teachers. really like them and are really happy. about them. I just always want more and. and I guess one of the things I would. like for you cubed. One of my personal. goals is that every teacher of maths. knows about you cubed. at the moment. Um a lot of teachers who. come to us are really happy they found. it but there's a lot of other teachers. who. don't know that it exists. I hope this. helps. Yeah. From a student perspective and not in. the classroom but at home studying.

and you know is there some um advice you. can give on how to best study. mathematics? So what's the role of the. student outside the classroom? Yeah, I think one thing we know is a lot. of people when they review material, whether it's maths or anything else, don't do it in the best way. I think a. problem a lot of people have is they. read through maybe a teacher's. explanation or a way of doing maths and.

it makes sense and they think, "Oh, yeah, I've got that." And they move on. Um, but then it's not until you come to. try and work on something and do a. problem that you actually realize you. didn't really understand it. just seem. to make sense. So I would say this is also something. that neuroscientists talk about to keep. giving yourself questions is a really. good way to study rather than looking. through lots of material. It's almost. like giving yourself lots of tests is a.

good way to actually deeply understand. things and know what you do when you. don't understand. So would the questions. be in the form of the material you're. reviewing is the answer to that question. or is it almost like beyond it's the. polygon thing you mentioned from a. square is it almost like I wonder. what is the bigger picture always kind. of asking of like how is this extended. and so on. Yeah, that that that would be. great. And it's a similar I mean a.

question I get asked a lot is about. homework. What is a good thing for kids. to do for homework? And one of the. recommendations I give is to not have. kids just do lots of questions for. homework, but to actually ask them to. reflect on what they've learned. Like. what was the big idea you were work you. learned today? Or what did you find. difficult? What did you struggle with? what was something that was exciting? Um, then kids go home and they have to.

kind of reflect in a deeper way. A lot. of times, I don't know if you had this. experience as a math student, lots of. people do. Kids are going through maths. questions, they're successful, they get. them right, but they don't even really. know what they're about. they and a lot. of kids go through many years of maths. like that doing lots of questions but. that really knowing what even the topic. is or what it's about what it's. important for so having students go back. and think at the end of a day what was. the big idea from this maths lesson why.

is it important where would I find that. in real life those are really good. questions for kids to be thinking about. it's probably for everybody to be. thinking about I think most Most of us. go through life never asking. like the bigger question. Almost like. you know those like layers of why. questions that kids ask when they're. very young. Yeah. We need to keep doing that. We like what uh that's the you know. whatever the term is you call first.

principles thinking. Some people call it. that. which is like. why are we doing it this way? So, one. one nice thing is to do that because. there's usually a good answer like the. reason we did it this way is because it. works for this reason. But then if you. want to do something totally novel is. you'll say well we've been doing it this. way. um because of historical reasons but. really this is not the best way to do.

it. there might be other ways and that's. how invention happens. right. and then you get you know that's really. useful in every aspect of life like. choosing your career choosing your um I. don't know what where you live. who your like romantic partner is like. everything. and I think it probably starts. [laughter] doing that in math class. that would be good if we started doing. that. I want I mean I wonder I I probably. didn't do very much of that for most of.

my education asking why except for later. much later in the subjects on like grad. school when you're doing research on. them when your first task of doing. something novel using this or solving a. problem really outside the classroom and. have to publish on is the first time you. think wait why are these things uh. interesting useful which are the things. that are useful and Yeah, I guess that.

was that would be nice if we did that. much earlier that u the quest of. invention. Yeah. Yeah. I mean, one of the sad. pieces of research data I think about is. the questions kids ask. um in school goes down like in in a. linear uh you know progression from in. the early years you can't stop kids. asking those questions but they learn. not to ask the questions. I think you. told somewhere about a early memory you.

had in your own education where you. asked the question or maybe that was an. example you gave but it was shut down. Oh yeah, [laughter] you've listened to. something I said. Yeah, I don't remember where it was. It caught. it caught me. Yeah, I remember it really vividly. Well, can you tell uh the memory? Yeah, I was. It's funny. I can remember. it must have really impacted me in that. moment because you know how there's lots. of hours of school you don't remember at. all but anyway um I I can remember where. I was sitting and everything I was in.

high a high school maths class although. they don't call it that in England and. um the teacher said and it was like the. first class of this teacher's class and. he said ask if you have any questions so. at one point I put my hand up and I said. I have a question and he said something. like that's your question. Um, [laughter]. and I was like, okay, I'm not asking any. more questions in this class. Hard in a way where you didn't want to. you the lesson you learned from that is.

I'm not going to ask. Yeah, that was absolutely the lesson I. that's the last question I'm asking. And. um that was yeah, he was the chair of. the maths department. I remember that. really well. So. maybe because of that experience, one of. the things we encourage when we teach. kids is asking questions and we value. when they ask questions and we put them. up on walls and celebrate and um. it's funny cuz I wish there was a.

feedback signal because he probably to. put a positive spin on it, he probably. didn't realize the negative impact he's. had in that moment, right? If he only. knew. See, this is probably when you're. more mature in grad school. I had an. amazing professor named Ali Shakafande. in computer science and he would get he. encouraged questions, but then he would. tell everybody how dumb their questions. are. But it was. it was done I guess if you show if you. say it with love and respect behind it,

then it's more like a friendly humorous. encouragement for more questions. It's an art, right? To write and. teaching is very hard. You have to time. it right because that's probably that. kind of humor is probably better for. when you're uh in grad school versus. when you're in the early education. Right. Well, and I guess kids or young. people get whether somebody's. doing it to be funny or you know has it. this I mean this is why teachers so.

hard. Even your tone can be impactful. It's so sad because like for. for that particular human the teacher. you could just had a bad day and one. statement can have a profound negative. impact. But I know sadly that maths there's a. lot of maths teachers who have that kind. of approach and they. I think they're suffering from the fact. that they think people are math people. are not math people and that comes. across in their teaching. But on the.

flip side, one positive statement to. keep them going. That's right. That is the flip side of. that. And I myself had like one teacher. who was really amazing for me in maths. and she kept me in the subject. Who was she? Who? She was um her name was Mrs. Marshall. [laughter]. and. um she was my A-level maths teacher. So. I was in. I mean. in England you do lots of subjects so.

you're 16 and then you choose like three. or four subjects. So I had chosen maths. and you go to higher levels. probably equivalent more to a master's. degree in the US cuz you're more. specialized. But anyways, she was my. teacher and for the first time in my. whole career in maths, she would give us. problems and tell us to talk about them. with each other. And so here I was. sitting there at like 17 talking with.

friends about how to solve a math. problem. And that was it. That was the. change that she made. But it was. profound for me. I because like those. calculus students, I started to hear. other people's ways of thinking and. seeing it and we would talk together and. come up with solutions and I was like. that was it that changed maths for me. and I so it wasn't some kind of personal. interaction with her. It was more like. she uh she was the catalyst for that. collaborative experience.

I mean yeah the many ways teachers can. inspire kids. I mean sometimes it's a. personal message but it can be your. teaching approach that changes maths for. kids you know uh Cal Newport. he uh he wrote a book called deep work. and he's a mathematician theoretical. computer scientist and he talks about. the kind of the focus required. to do that kind of workh. is there something you can comment on.

you know we live in a in a world full of. distractions. that that's seems like one of the. elements that makes studying and. especially the studying of subjects that. require thinking like math does. difficult. Is is there something from a student. perspective, from a teacher perspective. that encourages deep work. that you can comment? Yeah, I think. giving kids really inspiring deep. problems and we have some on our website.

is a really important experience for. them. Um, even if they only do it. occasionally, but it's really important. They actually. realize I I do I give a problem out. often when I'm working with teachers and. I say to them, "All right, I'm going to. check in with you after an hour." And. they were like, "An hour?" They think. it's shocking. And then they work on. this problem and after an hour I say, "Okay, how are we doing?" And they're. like, "An hour's gone by. [laughter].

How is this possible?" And. so everybody needs those like rich deep. problems. Most kids go through their. whole maths experience of however many. years, never once working on a problem. in that kind of deep way. So I the the. undergrad class I teach at Stanford, we. do that. We work on these deep problems. every session and the students come away. going, "Okay, I never want to go back to. that maths relationship I had where it.

was just all about quick answers. I I. just don't want to go back to that." And. so we can all all teachers can. incorporate those problems in their. classrooms. Maybe they don't do them. every day, but they at least give kids. some experience of being able to work. slowly and deeply and to go to deeper. places and not. be told they've got 5 minutes to finish. 20 questions. Yeah. Well, part of it is also just the.

um. the exercise of sitting there. maintaining focus for prolonged periods. of time. That's not often. I mean um that's a skill. Yeah, it's a skill that uh that also could be. discouraging like if you don't practice. it just sitting down for 10 minutes. straight and maintaining deep focus. could be exceptionally challenging like. if you're really thinking about a. problem. and to re I think it's really important. to realize that that's a skill that you.

can. just like a muscle you can build you can. start with 5 minutes and goes to 10. minutes to 30 and to an hour and and to. be successful I think in certain. subjects like mathematics you want to be. able to develop that skill. Otherwise, you're not going to. get to the the the the really rewarding. experience of solving these problems. Mhm. Mhm. Definitely. There was a survey. done of kids in school where they were. asked, "How long will you work on a. maths problem before you give up and. decide it's not possible to solve it?".

Question. And the result on average across the. kids was 2 minutes. [laughter]. Yeah. So that's a that's a bad sign, but. that's the it's a powerful sign that uh. they need to learn to not give up so. quickly. Yeah. Uh we mentioned offline. because we've been talking so much about. visualization. Uh Grant Sanderson, three Luan Brown. So.

he's inspired millions of people with. the kind of u exactly the kind of way of. thinking that you've been talking about. which is. yeah I love his work. converting sort of uh mathematical. concepts into visual. uh like uh visually representing them. exploring them in ways that uh help you. illuminate like the concepts. um what what do you think is the role of. that so he uses mostly programmatic. visualizations so it's the thing I. mention where there's like animations.

created by writing uh computer programs. Um like what what do you think how. scalable is that approach? But in. general, what do you think about his. approach? I think it's amazing. I should work with. him. [laughter]. I I can share some of our visuals and he. can make them in that amazing way. Um. so part of it is storytelling. Part of. it is like um [snorts] it's creating the. visuals and then weaving a story with. those visuals that kind of builds like.

there's also I mean there's also drama. in it. You start with a small example. and then you kind of all of a sudden. there's a surprise. Yeah. Yeah. [laughter] Yeah. And it really I mean it makes you fall. in love with the. That's right. with the concept. He he does talk about. that. his sense is like some of the stuff he. he he doesn't feel like he's teaching. Mhm. like the core curriculum which is. something you know he he sees himself as.

an inspirational figure but because I. think it's too difficult to kind of. convert all of the curriculum into those. elements. and and probably [clears throat] you. don't need to I mean you if people get. to experience mathematical ideas in the. way that he shares them um that will. change them and it will change the way. they they think and maybe they could go. on to take some other mathematical idea. and make it that beautiful. Well, he does that uh there's a he. created a library called Manom and he.

open sourced it and that library is the. uh people should check it out. It's. written in Python and he uses some of. those same elements like it allows you. to animate equations and animate little. shapes like people that you know he has. a very distinct style in his videos. And. what that resulted in, even though from. a software engineer perspective, the. code he released is not like super well. documented or perfect, but him releasing.

that. now there's all of these people. educating it. And the the cool to me. personally the coolest thing is to see. like. people they're not you know don't have. like a million subscribers or something. is they they they have just a few views. on the video but it just seems like the. process of them creating a video where. they teach is like transformative to. them from a student perspective. It's. the old Fineman thing. The best way to. learn is to teach, right?

And then him releasing that into the. wild is. Yeah. It it shows that that impact. Uh. yeah, absolutely. I think just giving. people that idea that you can do that. with maths and other subjects there. It's bound to be people all around who. can create more which is cool. Yeah, I definitely So I recommend people. do like JavaScript or Python. You can. you can build like visualizations of. most concepts in high school math. You. can do a lot of kinds of visualizations.

and doing that yourself. Plus, if you do. that yourself, people really love it. People actually. People love visualizations and math. Yeah. Because they I mean, it's something in. us that loves patterns, loves figuring. out difficult things and the patterns in. that are then are unexpected in some. way. Yeah. Have you ever noticed that hotels. are always filled with patterns? I was just noticing at the hotel I'm in. now. All of their carpets are pattern. carpets and then they have patterns on. the walls. Yeah.

So [laughter] yeah, we humans love the symmetry and. patterns, the breaking of symmetry and. patterns. Yeah. [laughter]. And it's funny that we don't see. mathematics as somehow intricately. connected to that, but it is right. I mean that's one of the perspectives I. love students to take is to be a pattern. seeker and. in [snorts] everything. in in yeah certainly in all of maths. I. mean you can think of all of maths as a. kind of subject of patterns and not just. visual patterns but you know when you.

think about multiplying by five and the. fact you can you know if you if you're. multiplying 18 * 5 you can instead think. of 9 * 10 that's a pattern that always. works in mathematics you can have a. number and double number and so yeah I. just think there are patterns everywhere. and if kids are thinking their role is. to see patterns and find patterns. It's. really exciting. What do you think about like MIT open.

courseware and the release of lectures. by universities? I think it's good. I think it's good. I. think the that is what started. the MOO I did was using that platform at. the. See, you ultimately think like the. Udacity model is a little bit more. effective than just a plain 2hour. lecture. I think there's definitely you. can bring in good pedagogy into online. learning and I think the idea of putting. things online so that people all over.

the world can access them is great. I. don't think the initial excitement. around MOOs sort of democratizing. education and making it more equal um. came about because they found that the. people taking MOUS tended to be the more. privileged people. Mhm. So that was I think there's still. something to be found in that. there's. still more to be done to help that. online learning reach those principles, but um definitely I think it's it's a.

good invention. And I have an online. class that's for kids that's little free. class that gives the topic. it's called how to learn maths. [clears throat]. How to learn math. Um it shows maths as. this visual creative subject and it. shares mindset and some brain science. and um kids who take it do better in. maths class. We've studied it with like. randomized control trials and given it. to middle school kids and other middle. school kids who don't take it but are.

taught by the same teachers. So their. teachers are the same and the kids who. take the online class end up 68% more. engaged in their maths class and do. better at the end of the year. So that's. a little six session 15minute class and. it changes kids maths relationship. So. it is true that we can do that with some. words. that aren't you know not it's not a huge. change to the education system.

Do you have advice for young people? We've been talking about mathematics. quite a bit, but uh in terms of their. journey through education, through their. career choices, through life, maybe. middle school, high school, undergrad. students of how to live a life they're. they can be proud of. I think if I were to give advice to. people, especially young people, my. advice would be to. always, it sounds really corny, but to.

always believe in yourself and know that. you can achieve. Because although that. sounds like obvious, of course, we want. kids to know that they can achieve. things. I know that millions of kids are. in the school system have been given the. message they cannot do things. And. adults, too. They have the idea, oh, I. did okay in this. I went into this job. because those other things I could never. have done okay in. So actually when they hear, hey, maybe. you could do those other things, even.

adults think, you know, maybe I can. And. they go back and they encounter this. knowledge and they relearn things and. they change careers and amazing things. happen. So for me, I think that message. is really important. You can learn. anything. Scientists try and find a. limit. They're always trying to find a. limit. Like how much can you really. learn? What's the limit to how much you. can learn? And they always come away not. being able to find it. People can just. go further and further and further. And. that is true of people born with brain.

um you know areas of their brain that. aren't functioning well that have what. we call special needs. Some of those. people also go on to develop and do. amazing things. So I think that really. experiencing that, knowing that, feeling, not just saying it, but knowing. it. deeply, you can learn anything is um um. something I wish all people would have. Actually also applies when you've.

achieved some level of success too. What. I find like in my life with people that. love me, when you achieve success, they. they keep celebrating your success and. they want you to keep doing the thing. that you were successful at. as opposed to believing in that you can. do the the something else, something. big, whatever your heart says to do. Right. And one of the things that I realized. the value of this. um you know quite recently which is sad. to say is how important it is to seek.

out when you're younger to seek out. mentors to seek out the. people like surround yourself with. people that will believe in you. Yeah. It's like a little bit is is on you. Mhm. It's like uh you don't get that. Um. sometimes if you go to like grad school, you think you kind of land on a mentor. Maybe you pick a mentor based on the. topic they're interested in. But the. reality is the people you surround. yourself with. They're going to define your life.

trajectory. So select. people. That's really true. And get away from. people who don't believe you. Sometimes. parents can be that they can love you. deeply but they be you know they set. it's the math thing we mentioned they. might set certain constraints on the. beliefs that you have and so in that if. you're interested in mathematics and. your parents are not that interested in. it don't listen to your parents on that. one dimension. Exactly.

Yeah. And if people tell you you can't. do things you have to hear from other. people who who believe in you. I think. you're absolutely right about that. So. sad the number of people who've had. those negative messages from parents. In. my limitless mind book, I interviewed. quite a few people who'd been told they. couldn't do maths. Sometimes by parents, sometimes by teachers and fortunately. they had got other ideas at some point. in their life and realized there was. this whole world of mathematical.

thinking that was open to them. [clears throat] So it's really important. that people do connect with people who. believe in them, however hard that might. be to find those people. What do you hope the education system, education in general looks like 10, 20, 50, 100 years from now? Are you. optimistic about this future? Yeah, I definitely have hope. There is. change can happen in the education. system. in recent years it's been. microscopically slow and um but.

I do actually see change happening like. we were talking earlier that data. science is now a course you can take in. high school instead of algebra 2 and. that's pretty amazing because that. content was set out in 1892 and hasn't. changed since then and so now we're. actually seeing a change in the content. event of high school. So I'm amazed that. that's happening and very happy it's.

happening. But so change is very slow in. education usually. But when you look. ahead. and think about all that we know and all. that we can offer kids in terms of. technology, you've got to think that a 100 years. from now, education will be totally. different to the way it is now. Maybe we. won't have subject boundaries anymore. because those don't really make much. sense. And. it's interesting to think how certain.

tools like programming maybe they'll be. deeply integrated in everything. You would think Yeah. You would think. that all kids are growing up learning to. program and create. So, um I just think. I mean the system of schooling we have. now people call it a factory model. It's. not designed to inspire creativity. And. I feel like that will also change. People might look back on these days and. think they were hilarious. But um maybe.

we'll in the future kids will be doing. their own programming and they'll be. able to learn things and find out things. and create things even as they're. learning. and um maybe the individual. subject boundaries will go. Data science. itself coming into the education system. kind of illustrates that because people. realize it doesn't really fit inside any. of the subjects. So what do we do with it? Where does it. go? And who teaches it? So it's already.

raising those kind of questions and. questioning how we have these different. subject boundaries. So you've seen data. science be integrated into the. curriculum. Yes, it's happening across the United. States as we speak. I wonder how that got initiated like how. does change happen in the education. system? Is it just a few revolutionary. like leaders? Does I think so? I think so. It's been. an interesting journey seeing data. science take off. Actually, it um there. was a.

course that was developed in 2014. by some people who thought data science. was a good idea for high schoolers and. then after some kids took the course and. nothing bad happened to them and they. went to college and people started to. accept it more and then this was a big. piece of the change in California. The. UC system communicated they sent out an. email last year to 50,000 high schools. saying we now accept data science. Kids.

can take it instead of algebra 2. That's a perfectly legitimate college. pathway. So that was like a big green. light for a lot of schools who are like. wondering about whether they could teach. it. So I think it happens in small. spaces and expands. So now. it goes viral. Yeah. In this modern age. then it goes viral. California's ahead. um I think in creating courses and. having kids go through it but it's. s c suddenly when I last looked there. were 12 states that were allowing data.

science as a high school course and I. think by next year that will have. doubled or more so a change is happening. Joe as I've said I think mathematics is. um is truly a beautiful subject and you. having an impact on millions of people's. lives by educating them, by inspiring. teachers to educate, in the ways that. you've talked about in multi-dimensional. ways, in visual ways. Um, I think it's.

incredible. So, you're you're spreading. beauty to the world [laughter] and so I. really really appreciate that you spend. your valuable time with me today. Thank. you for talking. Thank you. It was really good to talk to. you. Thanks for listening to this. conversation with Joe Bowler. To support. this podcast, please check out our. sponsors in the description. And now let. me leave you with some words from Albert. Einstein. Pure mathematics is the poetry of. logical ideas. Thanks for listening and hope to see you.

next time.