Transcript
00:00:00 The following is a conversation with Jo Bowler,
00:00:02 a mathematics educator at Stanford
00:00:04 and co founder of ucubed.org
00:00:08 that seeks to inspire young minds
00:00:10 with the beauty of mathematics.
00:00:12 To support this podcast,
00:00:14 please check out our sponsors in the description.
00:00:17 This is the Lex Friedman podcast
00:00:19 and here is my conversation with Jo Bowler.
00:00:23 What to you is beautiful about mathematics?
00:00:27 I love a mathematics that some people
00:00:30 don’t even think of as mathematics,
00:00:32 which is beautiful, creative mathematics,
00:00:37 where we look at maths in different ways,
00:00:39 we visualize it,
00:00:41 we think about different solutions to problems.
00:00:44 A lot of people think of maths
00:00:46 as you have one method and one answer.
00:00:49 And what I love about maths
00:00:51 is the multiple different ways you can see things,
00:00:54 different methods, different ways of seeing different.
00:00:56 In some cases, different solutions.
00:00:59 So that is what is beautiful to me about mathematics,
00:01:02 that you can see and solve it in many different ways.
00:01:05 And also the sad part that many people think
00:01:10 that maths is just one answer and one method.
00:01:13 So to you, the beauty emerges
00:01:16 when you have a problem with a solution
00:01:18 and you start adding other solutions,
00:01:21 simpler solutions, weirder solutions, more interesting,
00:01:25 some of their visual, some of their algebraic,
00:01:28 geometry, all that kind of stuff.
00:01:30 Yeah, I mean, I always say
00:01:32 that you can take any maths area and make it visual.
00:01:35 And we say to teachers,
00:01:36 give us your most dry, boring maths
00:01:39 and we’ll make it a visual, interesting, creative problem.
00:01:43 And it turns out you can do that with any area of maths.
00:01:46 And I think we’ve given,
00:01:49 it’s been a great disservice to kids and others
00:01:51 that it’s always been numbers, lots and lots of numbers.
00:01:56 Numbers can be great,
00:01:57 but you can think about maths in other ways besides numbers.
00:02:01 Do you find that most people are better visual learners
00:02:05 or is this just something that’s complimentary?
00:02:08 What’s the kind of the full spectrum of students
00:02:11 in the way they like to explore maths, would you say?
00:02:13 There’s definitely people who come into the classes I do
00:02:17 who are more interested in visual thinking
00:02:19 and like visual approaches.
00:02:21 But it turns out what the neuroscience is telling us
00:02:24 is that when we think about maths,
00:02:26 there are two visual pathways in the brain
00:02:29 and we should all be thinking about it visually.
00:02:32 Some approaches have been to say,
00:02:34 well, you’re a visual learner, so we’ll give you visuals
00:02:37 and you’re not a visual learner.
00:02:40 But actually, if you think you’re not a visual learner,
00:02:45 it’s probably more important that you have a visual approach.
00:02:48 So you can develop that part of your brain.
00:02:51 So you were saying that there’s some kind
00:02:52 of interconnected aspect to it.
00:02:54 So the visual connects with the non visual.
00:02:56 Yeah, so this is what the neuroscience has shown us
00:02:59 that when you work on a maths problem,
00:03:01 there are five different brain pathways
00:03:03 and that the most high achieving people in the world
00:03:06 are people who have more connections between these pathways.
00:03:10 So if you see a maths problem with numbers,
00:03:13 but you also see it visually,
00:03:15 that will cause a connection to happen in your brain
00:03:17 between these pathways.
00:03:19 And if you maybe write about it with words,
00:03:22 that would cause another connection
00:03:23 or maybe you build it with something physical
00:03:26 that would cause a different connection.
00:03:28 And what we want for kids is we call it
00:03:31 a multi dimensional experience of maths,
00:03:33 seeing it in different ways,
00:03:35 experiencing it in different ways,
00:03:37 that will cause that great connected brain.
00:03:40 You know, there’s these stories of physicists doing the same.
00:03:43 I find physicists are often better
00:03:45 at building that part of their brain
00:03:47 of using visualization for intuition building,
00:03:50 because you ultimately want to understand
00:03:53 the like the deepest secret underneath this problem.
00:03:57 And for that, you have to intuit your way there.
00:03:59 And you mentioned offline that one of the ways
00:04:02 you might approach a problem
00:04:03 is to try to tell a story about it.
00:04:05 And some of it is like legend,
00:04:07 but I’m sure it’s not always is, you know,
00:04:10 you have Einstein thinking about a train,
00:04:14 you know, and the speed of light
00:04:16 and you know, that kind of intuition is useful.
00:04:20 You start to like imagine a physical world,
00:04:22 like how does this idea manifest itself
00:04:25 in the physical world?
00:04:26 And then start playing in your mind
00:04:27 with that physical world and think,
00:04:29 is this going to be true?
00:04:30 Is this going to be true?
00:04:31 Right.
00:04:32 Einstein is well known for thinking visually.
00:04:35 And people talk about how he really didn’t want
00:04:39 to go anywhere with problems
00:04:41 without thinking about them visually.
00:04:43 But the other thing you mentioned
00:04:44 that sparked something for me is thinking with intuition,
00:04:48 like having intuition about math problems.
00:04:51 That’s another thing that’s often absent in math class,
00:04:54 the idea that you might think about a problem
00:04:55 and use your intuition, but so important.
00:05:00 And when mathematicians are interviewed,
00:05:02 they will very frequently talk about the role
00:05:04 of intuition in solving problems,
00:05:07 but not commonly acknowledged or brought into education.
00:05:12 Yeah, I mean, that’s what it is.
00:05:14 Like if you task yourself with building an intuition
00:05:18 about a problem, that’s where you start to pull in,
00:05:23 like what is the pattern I’m seeing?
00:05:26 In order to understand the pattern,
00:05:28 you might want to then start utilizing visualization.
00:05:31 But ultimately, that’s all in service
00:05:33 of like solving the puzzle, like cracking it open
00:05:38 to get the simple explanation of why things are wrong.
00:05:42 Why things are the way they are,
00:05:43 as opposed to, like you said, having a particular algorithm
00:05:47 that you can then execute to solve the problem.
00:05:50 Yeah, but it’s hard.
00:05:51 It’s hard, like reasoning is really hard.
00:05:53 Yeah, it’s hard.
00:05:55 I mean, I love to value what’s hard in maths
00:05:58 instead of being afraid of it.
00:06:00 We know that when you struggle,
00:06:02 that’s actually a really good time for your brain.
00:06:04 You want to be struggling when you’re thinking about things.
00:06:07 So if it’s hard to think intuitively about something,
00:06:10 that’s probably a really good time for your brain.
00:06:13 I used to work with somebody called Sebastian Thrun,
00:06:16 who is a great sort of mathematician,
00:06:19 you might think of him, AI person.
00:06:21 And I remember in one interview I did with him,
00:06:23 he talked about how they’d built robots,
00:06:25 I think for the Smithsonian,
00:06:27 and how they were having this trouble
00:06:29 with them picking up white noise.
00:06:32 And he said they had to solve it.
00:06:34 They had to work out what’s going on
00:06:36 and how he intuitively worked out what the problem was.
00:06:41 But then it took him three weeks to show it mathematically.
00:06:45 I thought that was really interesting
00:06:47 that how you can have this intuition
00:06:49 and know something works.
00:06:51 It’s kind of different from going through
00:06:53 that long mathematical process of proving it,
00:06:56 but so important.
00:06:58 Yeah, I think probably our brains are evolved
00:07:01 as like intuition machines
00:07:03 and the math of like showing it like formally
00:07:08 is probably an extra thing that we’re not designed for.
00:07:12 You see that with Feynman and his,
00:07:14 I mean, it just, all of these physicists,
00:07:16 definitely you see starting with intuition,
00:07:21 sometimes starting with an experiment
00:07:24 and then the experiment inspires intuition.
00:07:27 But you can think of an experiment
00:07:29 as a kind of visualization.
00:07:31 Just like let’s take whatever the heck we’re looking at
00:07:34 and draw it and draw like the pattern as it evolves,
00:07:38 as the thing grows for N equals one,
00:07:41 for N equals two, N equals three,
00:07:42 you start to play with it.
00:07:44 And then in the modern day, which I loved doing is,
00:07:49 you can write a program that then visualizes it for you.
00:07:52 And then you can start exploring it programmatically.
00:07:55 And then you can do so interactively too.
00:08:00 I tend to not like interactive
00:08:03 because it takes way too much work
00:08:05 because you have to click and move and stuff.
00:08:07 I love to interact through writing programs,
00:08:09 but that’s my particular brain, software engineer.
00:08:12 So like you can do all these kinds of visualizations
00:08:16 and then there’s the tools of visualization,
00:08:19 like color, all of those kinds of things
00:08:22 that you’re absolutely right.
00:08:24 They’re actually not taught very much.
00:08:27 Like the art of visualization.
00:08:28 Not taught.
00:08:30 And we love as well color coding.
00:08:33 Like when you represent something mathematically,
00:08:36 you can show color to show the growth and kind of code that.
00:08:41 So if I have an algebraic expression for a pattern,
00:08:44 maybe I show the X with a certain color,
00:08:46 but also write in that color
00:08:48 so you can see the relationship.
00:08:51 Very cool.
00:08:52 And yeah, particularly in our work
00:08:54 with elementary teachers,
00:08:56 many of them come to our workshops
00:08:58 and they’re literally in tears
00:09:00 when they see things making sense visually
00:09:03 because they’ve spent their whole lives
00:09:06 not realizing you can really understand things
00:09:09 with these visuals.
00:09:10 It’s quite powerful.
00:09:12 You say that there’s something valuable to learning
00:09:17 when the thing that you’re doing is challenging,
00:09:20 is difficult.
00:09:21 So a lot of people say math is hard
00:09:24 or math is too hard or too hard for me.
00:09:28 Do you think math should be easy or should it be hard?
00:09:33 I think it’s great when things are challenging,
00:09:36 but there’s something that’s really key
00:09:39 to being able to deal with challenging maths
00:09:42 and that is knowing that you can do it.
00:09:46 And I think the problem in education
00:09:48 is a lot of people have got this idea
00:09:51 that you’re either born with a maths brain or you’re not.
00:09:54 So when they start to struggle,
00:09:55 they think, oh, I don’t have that maths brain.
00:09:58 And then they will literally sort of switch off
00:10:00 in their brain and things will go downhill from that point.
00:10:04 So struggle becomes a lot easier
00:10:06 and you’re able to struggle if you don’t have that idea,
00:10:10 but you know that you can do it.
00:10:13 You have to go through this struggle to get there,
00:10:16 but you’re able to do that.
00:10:18 And so we’re hampered in being able to struggle
00:10:21 with these ideas we’ve been given about what we can do.
00:10:25 Can I ask a difficult question here?
00:10:26 Yeah.
00:10:27 So there’s kind of, I don’t know what the right term is,
00:10:31 but some people struggle with learning in different ways,
00:10:38 like their brain is constructed in different ways.
00:10:42 And how much should, as educators,
00:10:47 should we make room for that?
00:10:49 So how do you know the difference between this is hard
00:10:52 and I don’t like doing hard things
00:10:54 versus my brain is wired in a way
00:10:57 where I need to learn in very different ways.
00:10:59 I can’t learn it this way.
00:11:00 How do you find that line?
00:11:02 How do you operate in that gray area?
00:11:04 So this is why being a teacher is so hard
00:11:07 and people really don’t appreciate
00:11:09 how difficult teaching is when you’re faced with,
00:11:12 I don’t know, 30 students who think in different ways.
00:11:14 So, but this is also why I believe it’s so important
00:11:19 to have this multi dimensional approach to maths.
00:11:21 We’ve really offered it in one way,
00:11:24 which is here’s some numbers in a method.
00:11:27 You follow me, do what I just did and then reproduce it.
00:11:31 And so there are some kids who like doing that
00:11:33 and they do well.
00:11:34 And a lot of kids who don’t like doing it
00:11:37 and don’t do well.
00:11:39 But when you open up maths and you give,
00:11:42 you let kids experience it in different ways,
00:11:44 maybe visually with numbers, with words.
00:11:47 What happens is kids,
00:11:49 there are many more kids who can access it.
00:11:52 So those different brain wirings you’re talking about,
00:11:56 where some people are just more able to do something
00:11:58 in a particular way.
00:12:00 That’s why we want to,
00:12:01 that’s one of the reasons we want to open it up
00:12:04 so that there are different ways of accessing it.
00:12:08 And then that’s not really a problem.
00:12:11 So I grew up in the Soviet Union and fell in love
00:12:15 with math early.
00:12:17 I was forced into math early
00:12:19 and fell in love through force.
00:12:22 That’s good.
00:12:23 Well, good that you fell in love about the force.
00:12:25 Well, but something we talked about a little bit
00:12:28 is there is such a value for excellence.
00:12:33 It’s competitive and it’s also everybody kind of looks up
00:12:37 the definition of success is being in a particular class
00:12:44 is being really good at it.
00:12:47 And like, it’s not improving.
00:12:49 It’s like being really good.
00:12:50 I mean, we are much more like that with sports, for example.
00:12:53 We’re not, it’s like, it’s understood,
00:12:56 you’re going to star on the basketball team
00:12:59 if you’re gonna start on the basketball team
00:13:02 if you’re going to be better than the other guys,
00:13:05 the other girls on the team.
00:13:06 So that coupled with the belief,
00:13:12 this could be partially a communist belief, I don’t know,
00:13:14 but the belief that everybody is capable of being great.
00:13:19 But if you’re not great, that’s your fault
00:13:22 and you need to work harder.
00:13:23 And I remember I had a sense that probably delusional,
00:13:27 but I could win a Nobel prize.
00:13:29 I don’t even know what that entails.
00:13:31 But I thought, like my dad early on told me just offhand
00:13:39 and it always stuck with me that if you can figure out
00:13:42 how to build a time machine, how to travel back in time,
00:13:46 it will probably give you a Nobel prize.
00:13:48 And I remember early in my life thinking
00:13:50 I’m going to invent the time machine.
00:13:52 And like the tools of mathematics were in service
00:13:56 of that dream of winning the Nobel prize.
00:14:00 It’s silly. I didn’t really think in those concrete terms,
00:14:03 but I just thought I could be great at feeling.
00:14:06 And then when you struggle,
00:14:08 the belief that you could be great is like,
00:14:11 struggle is good.
00:14:12 Right, pushes you on, yeah.
00:14:14 And so the other thing about the Soviet system
00:14:17 that I’d love to hear your comments about
00:14:20 is just the sheer like hours of math.
00:14:23 Like the number of courses,
00:14:25 you’re talking about a lot of geometry, a lot more geometry.
00:14:29 I think in the American system,
00:14:30 you take maybe one year of geometry.
00:14:33 In high school, yeah.
00:14:33 In high school.
00:14:35 First of all, geometry is beautiful, it’s visual.
00:14:37 And then you get to reason through proofs
00:14:39 and stuff like that.
00:14:40 In Russia, I remember just being nailed
00:14:42 over and over with geometry.
00:14:43 It was just nonstop.
00:14:45 And then of course there’s different perspectives
00:14:48 on calculus and just the whole,
00:14:50 the sense was that math is like fundamental
00:14:56 to the development of the human mind.
00:14:58 So math, but also science and literature, by the way,
00:15:02 was also hit very hard.
00:15:04 Like we read a lot of serious adult stuff.
00:15:08 America does that a little bit too.
00:15:09 They challenge young adults with good literature,
00:15:12 but they don’t challenge adults very much with math.
00:15:16 So those two things, valuing excellence
00:15:20 and just a lot of math in the curriculum.
00:15:23 Do you think, do you find that interesting?
00:15:26 Because it seems to have been successful.
00:15:28 Yeah, I think that’s very interesting.
00:15:30 And there is a lot of success,
00:15:32 people coming through the Soviet system.
00:15:34 I think something that’s very different to the US
00:15:37 and other countries in the world
00:15:39 is that idea that excellence is important
00:15:41 and you can get there if you work hard.
00:15:44 In the US, there’s an idea that excellence is important,
00:15:48 but then kids are given the idea in many ways
00:15:52 that you can either do it
00:15:53 or you’re one of the people who can’t.
00:15:56 So many students in the school system
00:15:58 think they’re one of the kids who can’t.
00:15:59 So there’s no point in trying hard
00:16:02 because you’re never going to get there.
00:16:04 So if you can switch that idea, it would be huge.
00:16:09 And it seems from what you’ve said
00:16:11 that in the Soviet Union, that idea is really different.
00:16:16 Now, the downside of that idea that anybody can get there
00:16:19 if you work hard is that thought
00:16:24 that if you’re not getting there, it’s your fault.
00:16:26 And I would add something into that.
00:16:28 I would say that anybody can get there,
00:16:31 but they need to work hard and they also need good teaching
00:16:35 because there are some people who really can’t get there
00:16:37 because they’re not given access to that good teaching.
00:16:41 So, but that would be huge, that change.
00:16:44 As to doing lots of maths,
00:16:46 if maths was interesting and open and creative
00:16:50 and multi dimensional, I would be all for it.
00:16:53 We actually run summer camps at Stanford
00:16:56 where we invite kids in and we give them this maths
00:16:59 that I love.
00:17:00 And in our camp classrooms, they were three hours long.
00:17:05 And when we were planning, the teachers were like three hours,
00:17:09 are we going to be able to keep the kids excited
00:17:11 for three hours?
00:17:12 Turned out they didn’t want to go to break or lunch.
00:17:15 They’d be so into these mathematical patterns.
00:17:19 We couldn’t stop them.
00:17:21 It was amazing.
00:17:22 So yeah, if maths was more like that,
00:17:25 then I think having more of it would be a really good thing.
00:17:29 So what age are you talking about?
00:17:30 Is there, could you comment on what age is like
00:17:35 the most important when people quit math
00:17:39 or give up on themselves or on math in general?
00:17:42 And perhaps that age or something earlier
00:17:45 is really an important moment for them to discover,
00:17:48 to be inspired to discover the magic of math.
00:17:51 I think a lot of kids start to give up on themselves
00:17:54 and maths around from about fifth grade.
00:17:59 And then those middle school years are really important.
00:18:02 And fifth grade can be pivotal for kids
00:18:04 just because they’re allowed to explore
00:18:07 and think in good ways
00:18:10 in the early grades of elementary school.
00:18:12 But fifth grade teachers are often like,
00:18:13 okay, we’re going to prepare you now for middle school
00:18:15 and we’re going to give you grades and lots of tests.
00:18:18 And that’s when kids start to feel really badly
00:18:21 about themselves.
00:18:22 And so middle school years,
00:18:24 our camps are middle school students.
00:18:27 We think of those years as really pivotal.
00:18:29 Many kids in those years are deciding,
00:18:31 yes, I’m going to keep going with STEM subjects
00:18:35 or no, I’m not, that this isn’t for me.
00:18:38 So, I mean, all years are important
00:18:40 and in all years you can kind of switch kids
00:18:43 and get them on a different pathway.
00:18:45 But I think those middle school years are really important.
00:18:48 So what’s the role of the teacher in this?
00:18:50 So one is the explanation of the subject,
00:18:53 but do you think teachers should almost do like one on one,
00:18:58 you know, little Johnny, I believe in you kind of thing?
00:19:01 Like that energy of like.
00:19:03 Turns out it’s really important.
00:19:05 There’s a study that was done,
00:19:07 it was actually done in high school English classrooms
00:19:10 where all kids wrote an essay for their teacher.
00:19:14 And this was done as an experiment.
00:19:16 Half of the kids got feedback from their teacher,
00:19:18 diagnostic feedback, which is great.
00:19:21 But for half of the kids,
00:19:22 it said an extra sentence at the bottom
00:19:23 that the researchers had put on.
00:19:26 And the kids who read that extra sentence
00:19:30 did significantly better in English a whole year later.
00:19:33 The only change was this one sentence.
00:19:36 What did the sentence say?
00:19:37 So what did the sentence say?
00:19:39 The sentence said, I’m giving you this feedback
00:19:42 because I believe in you.
00:19:45 And the kids who read that did better a year later.
00:19:48 Yeah.
00:19:49 So when I share this with teachers,
00:19:52 I say, you know, I’m not suggesting
00:19:53 you put on the bottom of all kids work.
00:19:56 I’m giving this feedback because I believe in you.
00:19:58 One of the teachers said to me, we don’t put it on a stamp.
00:20:00 I said, no, don’t put it on a stamp.
00:20:03 It’s, but your words are really important.
00:20:07 And kids are sitting in classrooms all the time thinking,
00:20:12 what does my teacher think of me?
00:20:13 Does my teacher think I can do this?
00:20:17 So it turns out it is really important
00:20:19 to be saying to kids, I know you can do this.
00:20:22 And those messages are not given enough by teachers.
00:20:27 And really believe it.
00:20:28 And believe it.
00:20:29 Yeah, it’s like.
00:20:30 You can’t just say it, you have to believe it.
00:20:32 I sometimes, cause it’s like,
00:20:36 it’s such a funny dance,
00:20:38 cause I’m almost such a perfectionist.
00:20:39 I’m extremely self critical.
00:20:41 And I have one of the students come up to me
00:20:43 and it’s clear to me that they’re not even close to good.
00:20:49 And it’s tempting for me to be like,
00:20:52 to sort of give up on them mentally.
00:20:54 But the reality is like,
00:20:55 if you look at many great people throughout history,
00:20:58 they sucked at some point.
00:21:00 Yeah, exactly.
00:21:01 And some of the greatest took nonlinear paths
00:21:04 to where they sucked for long into later life.
00:21:08 And so always kind of believing that this person
00:21:11 can be great.
00:21:13 Exactly.
00:21:14 You have to communicate that,
00:21:15 plus the fact that they have to work hard.
00:21:17 That’s it, yeah.
00:21:19 Yeah, and you’re right.
00:21:19 Silicon Valley where I live is filled with people
00:21:22 who are dropouts at school,
00:21:24 or who had special needs, who didn’t succeed.
00:21:28 It’s very interesting that have gone on
00:21:30 to do amazing work in creative ways.
00:21:34 I mean, I do think our school system is set up
00:21:36 to value good memorizers who can reproduce
00:21:42 what a teacher is showing them,
00:21:44 and push away those creative deep thinkers,
00:21:49 often slower thinkers, they think slowly and deeply.
00:21:53 And they often get the idea early on
00:21:55 that they can’t be good at maths or other subjects.
00:21:59 So yeah, I think many of those people
00:22:02 are the ones who go on and do amazing things.
00:22:05 So there’s a guy named Eric Weinstein.
00:22:08 I know many mathematicians like this,
00:22:09 but he talks a lot about having a nonstandard way
00:22:16 of learning.
00:22:17 I mean, a lot of great mathematicians,
00:22:19 a lot of great physicists are like that.
00:22:22 And he felt like he became quickly,
00:22:24 he got his PhD at Harvard,
00:22:26 became quickly an outcast of the system.
00:22:28 Like the education, especially early education system,
00:22:32 didn’t help him.
00:22:33 Is there ways for an education system
00:22:37 to support people like that?
00:22:40 Is it this kind of multidimensional learning
00:22:42 that you mentioned?
00:22:43 Absolutely, absolutely.
00:22:44 I mean, I think education system still uses an approach
00:22:48 that was in classrooms hundreds of years ago.
00:22:50 The textbooks have a lot to answer for
00:22:53 in producing this very uninspiring mathematics.
00:22:58 But yeah, if you open up the subject
00:23:00 and have people see and solve it in different ways,
00:23:02 and value those different ways.
00:23:05 Somebody I appreciated a lot
00:23:07 is a mathematician called Mary Mizikani.
00:23:09 I don’t know if you’ve heard of her.
00:23:10 She won the Fields Medal.
00:23:12 She was from Iran.
00:23:14 First woman in the world
00:23:15 to win the Fields Medal in mathematics.
00:23:17 She died when she was 40.
00:23:19 She was at Stanford.
00:23:21 But her work was entirely visual.
00:23:24 And she talked about how her daughter
00:23:27 thought she was an artist
00:23:28 because she was always visualizing.
00:23:30 And I attended,
00:23:33 she asked me to chair the PhD defense
00:23:35 for one of her students.
00:23:37 And I went to the defense in the math department.
00:23:40 And it was so interesting
00:23:41 because this young woman spent like two hours
00:23:45 sharing her work.
00:23:46 All of it was visual.
00:23:47 In fact, I don’t think I saw any numbers at all.
00:23:50 That’s awesome.
00:23:51 And I remember that day thinking,
00:23:53 wow, I could have brought her like 13 year old
00:23:55 into this PhD defense.
00:23:56 They would not recognize this as maths.
00:24:00 But when Maryam Izzakhani won the Fields Medal,
00:24:03 all these other mathematicians were saying
00:24:05 that her work had connected
00:24:07 all these previously unconnected areas of maths.
00:24:11 And so, but when she was,
00:24:14 she also shared that when she was in school,
00:24:16 when she was about 13,
00:24:17 she was told that she couldn’t do maths.
00:24:21 She was told that by her teacher.
00:24:22 Is this is Iran?
00:24:23 She grew up in Iran.
00:24:24 In Iran, yeah.
00:24:26 So I love that.
00:24:27 To be told you can’t be good at maths
00:24:29 and then go on and win the Fields Medal is cool.
00:24:33 I’ve been told by a lot of people in my life
00:24:37 that I can’t do something.
00:24:38 I’m very, definitely nonstandard.
00:24:41 But all it takes,
00:24:42 that’s why people talk about like the one teacher
00:24:45 that changed everything.
00:24:46 That’s right.
00:24:47 All it takes is one teacher.
00:24:48 That’s right.
00:24:48 That’s the power of that.
00:24:53 So that’s like, that should be inspiring to teachers.
00:24:55 I think it is.
00:24:57 You as a single person, given the education system,
00:25:00 given the incentives,
00:25:00 you have the power to truly change lives.
00:25:03 And like 20 years from now,
00:25:04 I feel as a medalist will walk up to you
00:25:06 and say thank you.
00:25:07 Yeah, so you did that for me.
00:25:09 Yeah, absolutely.
00:25:10 And I share that with teachers
00:25:12 that even in this broken system
00:25:15 of what they have to do for districts and textbooks,
00:25:19 a single teacher can change kids maths relationship
00:25:22 or other subjects forever.
00:25:26 What’s the role of the parents in this picture?
00:25:29 Let’s go to another difficult subject.
00:25:31 Yeah, that is a difficult subject.
00:25:34 One study found that
00:25:38 the amount of maths anxiety parents had
00:25:42 predicted their child’s achievement in school,
00:25:46 but only if they helped with homework.
00:25:51 So…
00:25:52 Oh, that’s so funny.
00:25:57 There are some interesting implications for this.
00:25:59 I mean, you can see how it works.
00:26:01 If you have maths anxiety
00:26:02 and you’re helping your kids with homework,
00:26:03 you’re probably communicating things like,
00:26:06 oh, I was terrible at this at school.
00:26:09 And that’s how it gets passed on to kids.
00:26:12 So one implication is
00:26:14 if you have a really bad relationship with maths,
00:26:16 you hate maths, you have maths anxiety,
00:26:19 just don’t do maths homework with your kids.
00:26:22 But we have on our website,
00:26:25 we have a little sheet for parents
00:26:28 of ways to interact around maths with your kids and…
00:26:33 That’s ucubed.org.
00:26:36 That’s ucubed.org, yes.
00:26:39 So one of the things I say to parents
00:26:40 when I give parent presentations is
00:26:42 even if you hate maths,
00:26:44 you need to just fake it with your kids.
00:26:46 You should be always endlessly optimistic
00:26:49 and happy about doing maths.
00:26:52 And…
00:26:54 I’m always curious about this.
00:26:55 So I hope to have kids one day.
00:26:57 I don’t have kids currently.
00:27:00 Are parents okay with like sucking at maths
00:27:04 and then trying to get their kid
00:27:06 to be better than them essentially?
00:27:07 Like, is that difficult thing for a lot of parents?
00:27:09 It is difficult.
00:27:10 To have like, it’s almost like an ego thing.
00:27:12 Like, I never got good at this
00:27:15 and I probably should have.
00:27:16 And yeah, I mean, to me, you wanna celebrate that,
00:27:20 but I know a lot of people struggle with that.
00:27:23 Like coaches in sports,
00:27:25 to make an athlete become better than them,
00:27:29 it can be hard on the ego.
00:27:31 Yeah.
00:27:32 So do you experience the same with parents too?
00:27:35 I think, I mean, I haven’t experienced parents
00:27:38 worrying that their kids will be better than them.
00:27:40 I have experienced…
00:27:42 I have experienced parents
00:27:45 just having a really bad relationship with maths
00:27:47 and not wanting to help,
00:27:50 not knowing how to help, saying things.
00:27:54 Like another study showed that when mothers
00:27:56 say to their daughters,
00:27:58 I was bad at maths in school,
00:28:00 their daughter’s achievement goes down.
00:28:02 So we know that kids pick up on these messages
00:28:05 and which is why I say you should fake it.
00:28:08 But also I know that lots of people
00:28:10 have just had a really bad relationship with maths,
00:28:12 even successful people.
00:28:15 The undergrads I teach at Stanford
00:28:17 have pretty much always done well in maths,
00:28:21 but they come to Stanford thinking maths
00:28:24 is a set of methods to memorize.
00:28:28 And so, so do many parents believe that.
00:28:31 There’s one method that you memorize
00:28:34 and then you reproduce it.
00:28:36 So until people have really had an experience
00:28:40 of what I think of as the other maths,
00:28:42 well, until they’ve really seen
00:28:43 that it’s a really different subject,
00:28:48 it’s hard for them to be able to shift their kids
00:28:51 to see it differently.
00:28:52 Is there for a teacher,
00:28:55 if we were to like systematize it,
00:28:57 is there something teachers can do
00:28:59 to do this more effectively?
00:29:01 So you mentioned the textbook.
00:29:04 Yeah.
00:29:04 So what are the additional things
00:29:07 you can add on top of this whole old school
00:29:10 traditional way of teaching that can improve the process?
00:29:14 So I do think there’s a way of teaching maths
00:29:17 that changes everything for kids and teachers.
00:29:21 So I’m one of five writers of a new framework
00:29:25 for the state of California, a new maths framework.
00:29:27 It’s coming out next year.
00:29:29 And we are recommending through this maths framework
00:29:32 that people teach in this way.
00:29:33 It’s called teaching to big ideas.
00:29:37 So at the moment, people have standards
00:29:42 that have been written,
00:29:43 and then textbooks have taken these standards
00:29:45 and made not very good questions.
00:29:48 And if you look at the standards,
00:29:50 like I have some written down here,
00:29:53 just reading the standards,
00:29:54 it makes maths seem really boring and uninspiring.
00:29:57 What are the kind of, can you give a few examples?
00:30:00 So this is an interesting example.
00:30:03 In third grade, there are three different standards
00:30:06 about unit squares.
00:30:10 Okay.
00:30:11 So this is one of them.
00:30:12 A square with side length one unit, called a unit square,
00:30:16 is said to have one square unit of area
00:30:18 and can be used to measure area.
00:30:20 And that’s something you’re expected to learn?
00:30:23 That is something, so that’s a standard.
00:30:25 The textbook authors say,
00:30:26 oh, I’m gonna make a question about that.
00:30:28 And they translate the standards into narrow questions.
00:30:32 And then you measure success by your ability
00:30:33 to deliver on these standards.
00:30:36 So the standards themselves, I think of maths
00:30:40 and many people think of maths in this way
00:30:41 as a subject of like a few big ideas
00:30:44 and really important connections between them.
00:30:48 So you could think of it as like a network map
00:30:50 of ideas and connections.
00:30:52 And what standards do is they take that beautiful map
00:30:55 and they chop it up like this into lots of little pieces
00:30:58 and they deliver the pieces to schools.
00:31:01 And so teachers don’t see the connections between ideas,
00:31:05 nor do the kids.
00:31:06 So anyway, this is a bit of a long way of saying
00:31:08 that what we’ve done in this new initiative
00:31:11 is we have set out maths as a set of big ideas
00:31:16 and connections between them.
00:31:17 So this is grade three.
00:31:20 So instead of there being 60 standards,
00:31:25 we’ve said, well, you can pull these different standards
00:31:29 to get in with each other
00:31:32 and also value the ways these are connected.
00:31:38 And by the way, for people who are just listening,
00:31:39 we’re looking at a small number of like big concepts
00:31:44 within mathematics, square towels,
00:31:46 measuring fraction, shape and time,
00:31:49 and then how they’re interconnected.
00:31:51 And so the goal, this is for grade three, for example.
00:31:55 Yeah, and so we’ve set out for the state of California,
00:31:58 the whole of mathematics K10
00:32:02 as a set of big ideas and connections.
00:32:05 So we know that teachers, it works really well
00:32:08 if they say, okay, so a big idea in my grade is measuring.
00:32:15 And instead of reading five procedural statements
00:32:19 that involve measuring, they think,
00:32:21 okay, measuring is a big idea.
00:32:22 What rich, deep activity can I use
00:32:25 that teaches measuring to kids?
00:32:27 And as kids work on these deep, rich activities,
00:32:30 maybe over a few days,
00:32:32 turns out a lot of maths comes into it.
00:32:35 So we’re recommending that let’s not teach maths
00:32:40 according to all these multiple statements
00:32:43 and lots and lots of short questions.
00:32:45 Instead, let’s teach maths
00:32:47 by thinking about what are the big ideas
00:32:49 and what are really rich, deep activities
00:32:52 that teach those big ideas.
00:32:53 So that’s the, like how you teach it
00:32:55 and maximize learning.
00:32:57 What about like from a school district perspective,
00:33:01 like measuring how well you’re doing,
00:33:04 grades and tests and stuff like that.
00:33:07 Do you throw those out or is it possible?
00:33:08 I am not a fan of grades and tests myself.
00:33:13 I think grades are fine
00:33:15 if they’re used at the end of a course.
00:33:18 So at the end of my maths course,
00:33:21 I might get a grade because a grade is meant
00:33:23 to be a summative measure.
00:33:24 It kind of describes your summative achievement.
00:33:27 But the problem we have in maths classrooms
00:33:30 across the US is people use grades all the time,
00:33:33 every week or every day even.
00:33:36 My own kids, when they went through high school,
00:33:38 technology has not helped with this.
00:33:40 When they went through high school,
00:33:41 they knew they were being graded
00:33:42 for everything they did, everything.
00:33:45 And not only were they being graded for everything,
00:33:48 but they could see it in the grade book online
00:33:50 and it would alter every class they went into.
00:33:52 So this is the ultimate,
00:33:54 what I think of as a performance culture.
00:33:57 You’re there to perform, somebody’s measuring you,
00:34:00 you see your score.
00:34:03 So I think that’s not conducive for deep learning.
00:34:08 And yes, have a grade at the end of the year,
00:34:10 but during the year,
00:34:12 you can assess kids in much better ways.
00:34:14 Like teachers can, a great way of assessing kids
00:34:18 is to give them a rubric that kind of outlines
00:34:22 what they’re learning over the course of a unit
00:34:24 or a few weeks.
00:34:25 So kids can actually see the journey they’re on,
00:34:28 like this is what we’re doing mathematically.
00:34:31 Sometimes they self assess on those units
00:34:34 and then teachers will show what the kids can do
00:34:40 with a rubric and also write notes.
00:34:42 Like in the next few weeks,
00:34:43 you might like to learn to do this.
00:34:48 So instead of kids just thinking about,
00:34:49 I’m an A kid or a B kid,
00:34:51 or I have this letter attached to me,
00:34:53 they’re actually seeing mathematically what’s important
00:34:56 and they’re involved in the process
00:34:58 of knowing where they are mathematically.
00:35:02 At the end of the year, sure, they can have a grade,
00:35:04 but during the year,
00:35:06 they get these much more informative measures.
00:35:10 I do think this might be more for college,
00:35:14 but maybe not.
00:35:16 Some of the best classes I’ve had
00:35:18 is when I got a special like set aside,
00:35:23 like the professor clearly saw that I was interested
00:35:27 in some aspect of a thing.
00:35:29 And then I have a few in mind and one in particular,
00:35:34 when he said that he kind of challenged me.
00:35:38 So this is outside of grades and all that kind of stuff
00:35:41 that basically it’s like reverse psychology.
00:35:44 I don’t think this can be done.
00:35:47 And so I gave everything to do that particular thing.
00:35:50 So this happened to be in an artificial intelligence class.
00:35:54 But I think that like special treatment
00:35:57 of taking students who are especially like excellent
00:36:01 at a particular little aspect,
00:36:03 that you see their eyes light up.
00:36:06 I often think like maybe it’s tempting for a teacher
00:36:09 to think you’ve already succeeded there,
00:36:11 but they’re actually signaling to you
00:36:13 that like you could really launch them on their way.
00:36:18 And I don’t know, that’s too much to expect from teachers
00:36:21 I think to pay attention to all of that
00:36:24 because it’s really difficult.
00:36:25 But I just kind of remember who are the biggest,
00:36:30 the most important people in the history
00:36:32 of my life of education.
00:36:34 And it’s those people that who really didn’t just
00:36:37 like inspire me with their awesomeness, which they did,
00:36:40 but also just they pushed me a little.
00:36:42 Like they gave me a little push.
00:36:45 And that requires focusing on the quote unquote
00:36:47 excellent students in the class.
00:36:50 Yeah, I think what’s important though is teachers
00:36:53 to have the perspective that they don’t know
00:36:56 who’s gonna be excellent at something
00:36:58 before they give out the activity.
00:37:00 Exactly.
00:37:01 And in our camp classes that we ran,
00:37:05 sometimes students would finish ahead of other students.
00:37:10 And we would say to them, can you write a question
00:37:15 that’s like this but different?
00:37:19 Oh, and over time we encourage them
00:37:21 to like extend things further.
00:37:24 I remember we were doing one activity
00:37:26 where kids were working out the borders of a square
00:37:30 and how big this border would be in different case sizes.
00:37:33 And one of the boys came up at the end of the class
00:37:36 and said, I’ve been thinking about how you do this
00:37:38 to the Pentagon.
00:37:40 And I said, that’s fantastic.
00:37:42 How do you, what does it look like with Pentagon?
00:37:44 Go find out, see if you can discover.
00:37:47 So I didn’t know he was gonna come up and say that.
00:37:49 And I didn’t have in my head like this is the kid
00:37:53 who could have this extension task,
00:37:55 but you can still do that as a teacher.
00:37:59 When kids get excited about something
00:38:01 or they’re doing well in something,
00:38:03 have them extend it, go further.
00:38:06 It’s great.
00:38:07 And then you also, like this is like teacher and coach,
00:38:11 you could say it in different ways to different students.
00:38:13 Like for me, the right thing to say is almost to say,
00:38:18 I don’t think you could do this, this is too hard.
00:38:21 Like that’s what I need to hear.
00:38:23 It’s just like, no, there’s immediate push.
00:38:26 But with some people, if they’re a little bit more,
00:38:29 I mean, it’s all has to do with upbringing,
00:38:31 how your genetics is.
00:38:33 They might be much more, that might break them.
00:38:35 Yeah, that might break them.
00:38:36 And so you have to be also sensitive to that.
00:38:38 I mean, teaching is really difficult for this very reason.
00:38:42 It is.
00:38:43 So what is the best way to teach math,
00:38:47 to learn math at those early few days
00:38:50 when you just wanna capture them?
00:38:53 I do something, actually there’s a video of me
00:38:56 doing this on our website that I love
00:38:58 when I first meet students.
00:39:03 And this is what I do.
00:39:04 I show them a picture, this is the picture I show them.
00:39:08 And it’s a picture of seven dots like this.
00:39:12 And I show it for just a few seconds and I say to them,
00:39:15 I’d like you to tell me how many dots there are,
00:39:17 but I don’t want you to count them.
00:39:19 I want you to group the dots.
00:39:20 And I show it them and then I take it away
00:39:24 before they’ve even had enough time to count them.
00:39:27 And then I ask them, so how did you see it?
00:39:29 And I go around the room and amazingly enough,
00:39:35 there’s probably 18 different ways
00:39:36 of seeing these seven dots.
00:39:39 And so I ask people, tell me how you grouped it.
00:39:42 And some people see it as like an outside hole
00:39:44 with a center dot.
00:39:45 Some people see like stripes of lines.
00:39:49 Some people see segments.
00:39:51 And I collect them all and I put them on the board.
00:39:53 And at the end I say, look at this,
00:39:55 we are a class of 30 kids
00:39:57 and we saw these seven dots in 18 different ways.
00:40:00 There’s actually a mathematical term for this.
00:40:01 It’s called groupitizing.
00:40:03 Groupitizing?
00:40:04 Yeah.
00:40:05 I like it.
00:40:06 It’s kind of cool.
00:40:07 So turns out though that how well you groupitize
00:40:11 predicts how well you do in maths.
00:40:15 Is it a raw talent or is it just something
00:40:18 that you can develop?
00:40:19 I don’t think it’s raw.
00:40:20 I don’t think you’re born groupitizing, I think,
00:40:22 but some kids have developed that ability if you like.
00:40:27 And you can learn it.
00:40:28 So this to me is part of how wrong we have maths.
00:40:34 That we think to tell whether a kid’s good at maths,
00:40:36 we’re gonna give them a speed test on multiples.
00:40:40 But actually seeing how kids group dots
00:40:43 could be a more important assessment
00:40:46 of how well they’re gonna do in maths.
00:40:48 Anyway, I diverge.
00:40:49 What I like to do though when I start off with kids
00:40:52 is show them I’m gonna give you maths problems.
00:40:54 I’m gonna value the different ways you see them.
00:40:57 And it turns out you can do this kind of problem
00:40:59 asking people how they group dots with young children
00:41:03 or with graduate students.
00:41:05 And it’s engaging for all of them.
00:41:10 You talk about creativity a little bit
00:41:12 and flexibility in your book Limitless.
00:41:15 What’s the role of that?
00:41:16 So it sounds like there’s a bit of that kind of thing
00:41:19 involved in groupitizing.
00:41:22 Yeah, yeah.
00:41:24 I love this term.
00:41:25 So what would you say is the role of creativity
00:41:28 and flexibility in the learning of maths?
00:41:31 I think what we know now is that what we need
00:41:34 for this 21st century world we live in is a flexible mind.
00:41:42 School should not really be about teaching kids
00:41:44 particular methods but teaching them
00:41:46 to approach problems with flexibility.
00:41:49 Being creative, thinking creatively is really important.
00:41:52 So people don’t think the words maths and creativity
00:41:56 come together, but that’s what I love about maths
00:42:00 is the creative different ways you can see it.
00:42:02 And so helping our kids, there’s a book I like a lot
00:42:08 by physicists, you probably know this book called Elastic.
00:42:13 You might know it.
00:42:15 And it’s about how we want elastic minds.
00:42:18 Same kind of thing, flexible, creative minds.
00:42:21 And schools do very little on developing that kind of mind.
00:42:26 They do a lot of developing the kind of mind
00:42:31 that a computer now does for us.
00:42:35 Memorization.
00:42:36 Memorization, doing procedures, a lot of things
00:42:41 that we spend a lot of time in school on.
00:42:45 In the world, when kids leave school,
00:42:47 a computer will do that and better than they will.
00:42:52 But that creative, flexible thinking,
00:42:54 we’re kind of at ground zero at computers
00:42:57 being able to engage in that thinking.
00:42:59 Maybe we’re a little above ground zero,
00:43:00 but the human brain is perfectly suited
00:43:05 for that creative, flexible thinking.
00:43:07 That’s what humans are so great at.
00:43:09 So I would like the balance to shift in schools.
00:43:12 Maybe you still need to do some procedural kind of thinking,
00:43:15 but there should be a lot more of that creative,
00:43:17 flexible thinking.
00:43:20 And what’s the role of other humans in this picture?
00:43:24 So collaborative learning, so brainstorming together.
00:43:29 So creativity as it emerges from the collective intelligence
00:43:33 of multiple humans.
00:43:34 Yeah, super important.
00:43:36 And we know that also helps develop your brain,
00:43:40 that social side of thinking.
00:43:42 And I love mathematics collaboration
00:43:44 where people build on each other’s ideas
00:43:47 and they come up with amazing things.
00:43:49 I actually taught a hundred students calculus
00:43:54 at Stanford recently, undergrads,
00:43:56 and we taught them to collaborate.
00:43:59 So these students came in Stanford
00:44:01 and most of them were against collaboration in math.
00:44:04 This is before COVID in person?
00:44:06 Yeah, it was just before COVID hit.
00:44:08 It was 2019.
00:44:10 And this summer.
00:44:12 So you said they’re against?
00:44:13 Yeah, so it’s really interesting.
00:44:16 So they’d only experienced maths individually
00:44:20 in a kind of competitive individual way.
00:44:22 And if they had experienced it as group work,
00:44:24 it had been a bad experience.
00:44:26 Like maybe they were the one who did it all
00:44:28 and the others didn’t do much.
00:44:31 So they were kind of against collaboration.
00:44:33 They didn’t see any role for it in maths.
00:44:35 And we taught them to collaborate and it was hard work
00:44:39 because as well as the fact
00:44:42 that they were kind of against collaboration,
00:44:43 they came in with a lot of like social comparison thinking.
00:44:48 So I’m in this room with other Stanford undergrads
00:44:51 and they’re better than me or…
00:44:53 So when we set them to work on a maths problem together,
00:44:55 the first one was kind of a disaster
00:44:57 because they put all like, they’re better than me.
00:44:59 They’re faster than me.
00:45:00 They came up with something I didn’t come up with.
00:45:02 So we taught them to let go of that thinking
00:45:05 and to work well together.
00:45:08 And one of the things we did, we decided
00:45:10 we wanted to do a pre and post test
00:45:12 at the end of this teaching.
00:45:13 It was only four weeks long, but we knew
00:45:16 we didn’t want to give them like a time test
00:45:18 of individual work.
00:45:19 So we gave them an applied problem to do at the beginning
00:45:23 and we gave them to do in pairs together.
00:45:26 And we gave each of them a different colored pen
00:45:28 and said, work on this activity together
00:45:30 and keep using that pen.
00:45:33 So then we had all these pieces of student work.
00:45:35 And what we saw was they just worked
00:45:37 on separate parts of the paper.
00:45:39 So there’s a little like red pen section
00:45:42 and a green pen section.
00:45:43 And they didn’t do that well on it.
00:45:45 Even though it was a problem that middle
00:45:48 or high school kids could do,
00:45:49 but it was like a problem solving kind of problem.
00:45:52 And then we gave them the same one to do at the end,
00:45:55 gave them the same colors.
00:45:56 And actually they had learned to collaborate.
00:45:59 And not only were they collaborating the second time round,
00:46:02 but that boosted their achievement.
00:46:04 And the ones who collaborated did better on the problem.
00:46:07 Collaboration is important, having people,
00:46:10 and what was so eye opening for these undergrads
00:46:12 and they talked about it in lovely ways
00:46:15 was I learned to value other people’s thinking on a problem.
00:46:19 And I learned to value that other people
00:46:22 saw it in different ways.
00:46:24 And it was quite a big experience for them
00:46:29 that they came out thinking,
00:46:31 I can do maths with other people.
00:46:33 People can see it differently.
00:46:34 We can build on each other’s ways of thinking.
00:46:38 I got a chance to,
00:46:39 I don’t know if you know who Daniel Kahneman is,
00:46:42 got a chance to interact with him.
00:46:44 And like the first,
00:46:45 cause he had a few,
00:46:46 but one famous collaboration throughout his life
00:46:51 with Tversky.
00:46:52 And just like, you know,
00:46:55 he hasn’t met me before in person,
00:46:58 but just the number of questions he was asking,
00:47:01 just the curiosity.
00:47:03 So I think one of the skills,
00:47:05 the collaboration itself is a skill.
00:47:07 And I remember my experience with him was like,
00:47:10 okay, I get why you’re so good at collaboration
00:47:14 because he was just extremely good at listening
00:47:17 and genuine curiosity about how the other person
00:47:21 thinks about the world, sees the world.
00:47:23 And then together he’s,
00:47:24 he pulled me in in that particular case.
00:47:27 He doesn’t know in particular,
00:47:29 like that much about autonomous vehicles,
00:47:31 but he kept like asking all of these questions.
00:47:34 And then like 10 minutes in,
00:47:36 we’re together trying to solve the problem
00:47:38 of autonomous driving.
00:47:39 And like, and that, I mean, that’s really fulfilling.
00:47:43 That’s really enriching.
00:47:43 But it also in that moment made me realize
00:47:46 it’s kind of a skill.
00:47:48 Cause you have to kind of put your ego aside,
00:47:50 put your view of the world aside
00:47:52 and try to learn how the other person sees it.
00:47:55 And the other thing you have to put aside
00:47:57 is this social comparison thinking.
00:48:00 Like if you are sitting there thinking,
00:48:03 wow, that was an amazing idea.
00:48:04 He’s so much better than I am.
00:48:06 That’s really gonna stop you taking on
00:48:09 the value of that idea.
00:48:12 And so there’s a lot of that going on
00:48:14 between these Stanford students when they came.
00:48:17 And trying to help them let go of that.
00:48:21 One of the things I’ve discovered
00:48:24 just because being a little bit more in the public eye,
00:48:27 how rewarding it is to celebrate others.
00:48:31 And how much is going to actually pay off in the long term.
00:48:35 So this kind of silo thinking of like,
00:48:38 I want to prove to a small set of people around me
00:48:41 that I’m really smart and do so
00:48:45 by basically not celebrating
00:48:47 how smart the other people are.
00:48:48 That’s actually maybe short term,
00:48:51 it seems like a good strategy, but long term it’s not.
00:48:54 And I think if you practice at the student level
00:48:56 and then at the career level, at every single stage,
00:48:58 I think that’s ultimately.
00:49:00 I agree with you.
00:49:01 I think that’s a really good way of thinking about it.
00:49:04 You mentioned textbooks and you didn’t say it,
00:49:11 maybe textbooks isn’t the perfect way to teach mathematics,
00:49:15 but I love textbooks.
00:49:16 They’re like pretty pictures
00:49:18 and they smell nice and they open.
00:49:19 I mean, I talk about like physical.
00:49:21 Some of my greatest experiences have been just like,
00:49:25 cause they’re really well done.
00:49:27 When we’re talking about basic, like high school,
00:49:30 calculus, biology, chemistry, those are like,
00:49:34 those are incredible.
00:49:36 It’s like Wikipedia, but with color and a nice little.
00:49:40 You must’ve seen some good textbooks
00:49:41 if they had pretty pictures and color.
00:49:44 Yeah, I mean, I remember,
00:49:46 I guess it was very, very standard, like AP calculus,
00:49:51 AP biology, AP chemistry.
00:49:53 I felt those were like some of the happiest days of my life
00:49:56 in terms of learning was high school.
00:49:58 Cause it was very easy, honestly.
00:50:01 It felt hard at the time,
00:50:03 but you’re basically doing a whirlwind tour
00:50:07 of all the science.
00:50:08 Yeah, yeah.
00:50:09 Without having to pick, you do literature,
00:50:12 you do like Shakespeare, calculus, biology, physics,
00:50:17 chemistry, what else?
00:50:19 Anatomy, physiology, computer science,
00:50:23 without like, nobody’s telling you what to do with your life.
00:50:26 You’re just doing all of those things.
00:50:28 That’s a good thing, you’re right.
00:50:29 But I remember the textbooks weren’t,
00:50:32 I mean, maybe I’m romanticizing the past,
00:50:35 but I remember they weren’t, they’re pretty good.
00:50:38 But so you think, what role do you think they play still?
00:50:42 And like in this more modern digital age,
00:50:45 what’s the best materials with which to do
00:50:48 these kinds of educations?
00:50:50 Well, I’m intrigued that you had such a good experience
00:50:52 with textbooks.
00:50:53 I mean, I can remember loving some textbooks I had
00:50:57 when I was learning and I love books.
00:50:59 I love to pick up books and look through them,
00:51:03 but a lot of maths textbooks
00:51:06 are not good experiences for kids.
00:51:08 They, we have a video on our website
00:51:12 of the kids who came to our camp
00:51:14 and one of the students says,
00:51:15 in maths, you have to follow the textbook.
00:51:18 The textbook is kind of like the Bible.
00:51:20 You have to follow it.
00:51:22 And every day it’s slightly different.
00:51:25 Like on Monday you do 2.3.2
00:51:28 and on Tuesday you do 2.3.3 and on Wednesday.
00:51:32 And you never go off that.
00:51:35 That’s like every single day.
00:51:37 And that’s not inspiring for a lot of the kids.
00:51:43 So one of the things they loved about our camp
00:51:45 was just that there were no books.
00:51:46 Even though we gave them sheets of paper instead,
00:51:50 they still felt more free
00:51:53 because they weren’t just like trotting through exercises,
00:51:56 exercises, so.
00:51:59 Like what a textbook allows you is like you’re,
00:52:03 the very thing you said they might not like,
00:52:06 the 2.3, 2.3, it feels like you’re making progress.
00:52:11 And like it’s little celebrations
00:52:12 because you do the problem and it seems really hard
00:52:16 and you don’t know how to do it.
00:52:17 And then you try and try and then eventually succeed.
00:52:20 And then you make that little step and further progress.
00:52:23 And then you get to the end of a chapter
00:52:26 and you get to like, it’s closure.
00:52:28 You’re like, all right, I got that figured out.
00:52:29 And then you go on to the next chapter.
00:52:31 I can see that.
00:52:32 I mean, I think it could be in a textbook.
00:52:34 You can have a good experience with a textbook,
00:52:38 but what’s really important is what is in that textbook?
00:52:43 What are you doing inside it?
00:52:45 And I mean, I grew up in England
00:52:47 and in England we learn maths.
00:52:50 We don’t have this separation of algebra and geometry.
00:52:54 And I don’t think any other country
00:52:56 apart from the US has that.
00:52:58 But I look at kids in algebra classes
00:53:01 where they’re doing algebra for a year.
00:53:02 And I think I would have been pretty bored doing that.
00:53:05 By the way, can we analyze your upbringing real quick?
00:53:13 Why do British folks call mathematics, maths?
00:53:19 Why is it the plural?
00:53:21 Is it because of everything you’re saying
00:53:23 where it’s a bunch of subdisciplines?
00:53:25 Yeah, I mean, mathematics is supposed to be
00:53:29 the different maths that you look at,
00:53:34 whether you think of that as topics
00:53:38 like geometry and probability,
00:53:39 or I think of it as maths is just multi dimensional
00:53:45 lots of ways, but that’s why it was called mathematics.
00:53:48 And then it was shortened to maths.
00:53:50 And then for some reason it was just math in the US.
00:53:53 But to me, math has that more singular feel to it.
00:53:58 And there’s an expression here, which is do the math,
00:54:03 which basically means do a calculation.
00:54:05 That’s what people mean by do the math.
00:54:07 So I don’t like that expression
00:54:09 because no math could be anything
00:54:11 doesn’t have to be calculation.
00:54:12 And so yeah, I like maths
00:54:14 because it has more of that broad feel to it.
00:54:17 Yeah, I love that.
00:54:18 Maths kind of emphasize the multi dimensional,
00:54:20 like a variety of different subdisciplines,
00:54:22 different approaches.
00:54:23 Yeah.
00:54:25 Okay, but outside of the textbook,
00:54:28 what do you see like broadly being used?
00:54:33 You mentioned Sebastian Thrun and MOOCs,
00:54:36 online education.
00:54:37 Do you think that’s an effective set?
00:54:39 Can be.
00:54:40 I mean, online, having great teachers online,
00:54:45 obviously extends those teachers to many more people.
00:54:48 And that’s a wonderful thing.
00:54:51 I have quite a few online courses myself.
00:54:54 I got the bug working with Sebastian
00:54:56 when he had released his first MOOC.
00:54:59 And I thought, maybe I could do one in maths education.
00:55:03 And I didn’t know if anybody would take it.
00:55:06 I remember releasing it that first summer
00:55:08 and it was a free online class
00:55:09 and 30,000 maths teachers took it that first summer.
00:55:13 And they were all talking about it with each other
00:55:16 and sharing it and it was like took off.
00:55:18 In fact, it was that MOOC that got me to create YouCubed
00:55:23 with Kathy Williams, who’s the co founder,
00:55:26 because people took the MOOC and then they said,
00:55:28 okay, what now?
00:55:30 I finished, what can I have next?
00:55:33 And so that was where we made our website.
00:55:35 But so yeah, I think online education can be great.
00:55:39 I do think a lot of the MOOCs don’t have great pedagogy.
00:55:43 They’re just a talking head.
00:55:45 And you can actually engage people in more active ways,
00:55:50 even in online learning.
00:55:53 So I learned from the Udacity principle
00:55:55 when I was working at Udacity,
00:55:57 never to talk more than like five minutes.
00:56:00 And then to ask people to do something.
00:56:03 So that’s the sort of pedagogy of the online classes I have
00:56:07 is a little bit of presenting something
00:56:10 and then people do something and there’s a little bit more.
00:56:13 Because I think if you have a half hour video,
00:56:15 you just switch off and start doing other things.
00:56:19 So the way Udacity did it is like five, 10 minute,
00:56:22 like bit of teaching with some visual stuff perhaps.
00:56:27 And then there’s like a quiz almost.
00:56:29 Then you answer a question, yeah.
00:56:30 Yeah, no, that’s really effective.
00:56:34 You mentioned Ucubed, so what’s the mission?
00:56:36 What’s the goal?
00:56:36 You mentioned how it started, but what’s yeah,
00:56:41 where are you at now?
00:56:42 And what’s your dream with it?
00:56:44 What are the kind of things that people should go
00:56:47 and check out?
00:56:47 Yeah, we started Ucubed, I guess it was about five years
00:56:51 ago now and we’ve had over 52 million visitors to the site.
00:56:55 So I’m very happy about that.
00:56:57 And our goal is to share good ideas for teaching
00:57:01 with teachers, students, parents in maths and to help.
00:57:07 We have a sort of sub goal of a raising maths anxiety.
00:57:10 That’s important to us, but also to share maths
00:57:13 as this beautiful creative subject.
00:57:15 And it’s been really great.
00:57:19 We have lessons on the site, but one of the reasons
00:57:23 I thought this was needed is there’s a lot of knowledge
00:57:26 in the academy about how to teach maths well.
00:57:29 Loads and loads of research and journals
00:57:32 and lots of things written up, but teachers don’t read it.
00:57:36 They don’t have access to it.
00:57:39 They’re often behind pay walls, they’ve written
00:57:43 in really inaccessible ways.
00:57:46 So people wouldn’t want to read them or understand them.
00:57:49 So this actually is a big problem.
00:57:51 You have this whole industry of people finding out
00:57:54 how to teach well, not sharing it with the people
00:57:57 who are teaching.
00:57:58 So that’s why we made Ucubed.
00:58:01 And instead of just putting articles up saying,
00:58:03 here’s some things to read about how to teach well,
00:58:05 we translated what was coming from research
00:58:08 into things that teacher could use.
00:58:10 So lessons, there were videos to show kids
00:58:13 and there were tips for parents.
00:58:16 There were all sorts of things on the site.
00:58:17 And it’s been amazing.
00:58:20 As we took inspiration from the week of code,
00:58:24 which got teachers to focus on coding for a week.
00:58:29 And we have this thing called the week
00:58:31 of inspirational maths.
00:58:33 And we say, just try it for a week.
00:58:35 Just give us one week and try it and see what happens.
00:58:39 And so it’s been downloaded millions of times.
00:58:42 Teachers use it every year.
00:58:44 They start the school year with it.
00:58:46 And what they tell us is it was amazing.
00:58:49 The kids lights were on, they were excited, they loved it.
00:58:52 And then the week finished and I opened my textbooks
00:58:56 and the lights went out and they were not interested.
00:59:00 Yeah, but getting that first inspiration is still powerful.
00:59:04 It is, I wish, I mean, what I would love
00:59:07 is if we could actually extend that for the whole year.
00:59:11 We’re a small team at Stanford
00:59:13 and we’re trying to keep up with great things
00:59:17 to put on the site.
00:59:19 We haven’t the capacity to produce
00:59:22 these creative visual maths tasks
00:59:24 for every year group for every day,
00:59:25 but I would love to do that.
00:59:27 How difficult is it to do?
00:59:28 I mean, it’s to come up with visual formulations
00:59:33 of these big important topics you need to think about
00:59:39 in a way that you could teach.
00:59:43 I mean, we can do it.
00:59:44 We actually, we went from the week of inspirational maths
00:59:47 and we made K8 maths books with exactly that.
00:59:51 Big ideas, rich activities, visuals.
00:59:54 We just finished the last one.
00:59:56 We’ve been doing it for five years
00:59:58 and it’s been exhausting and we just finished.
01:00:01 So now there’s a whole K8 set of books
01:00:05 and they’re organized in that way.
01:00:07 These are the big ideas.
01:00:08 Here are rich, deep activities.
01:00:11 They’re not though what you can do every day for a year.
01:00:16 So some teachers use them as a kind of supplement
01:00:19 to their boring textbook.
01:00:21 And some people have said, okay, this is the year.
01:00:24 This book tells us what the year is
01:00:26 and then we’ll supplement these big activities with.
01:00:29 So they’re being used and teachers really like them
01:00:34 and are really happy about them.
01:00:36 I just always want more.
01:00:37 And I guess one of the things I would like for YouCubed,
01:00:40 one of my personal goals is that every teacher of maths
01:00:43 knows about YouCubed.
01:00:45 At the moment, lots of teachers who come to us
01:00:48 are really happy they found it
01:00:49 but there’s a lot of other teachers
01:00:51 who don’t know that it exists.
01:00:53 I hope this helps.
01:00:54 Yeah.
01:00:55 From a student perspective and not in the classroom
01:00:59 but at home studying,
01:01:03 is there some advice you can give
01:01:07 on how to best study mathematics?
01:01:09 So what’s the role of the student outside the classroom?
01:01:13 Yeah, I think one thing we know is a lot of people
01:01:17 when they review material,
01:01:19 whether it’s maths or anything else,
01:01:21 don’t do it in the best way.
01:01:23 I think a problem a lot of people have
01:01:25 is they read through maybe a teacher’s explanation
01:01:28 or a way of doing maths and it makes sense.
01:01:32 And they think, oh yeah, I’ve got that.
01:01:33 And they move on.
01:01:36 But then it’s not until you come to try
01:01:38 and work on something and do a problem
01:01:40 that you actually realize you didn’t really understand it,
01:01:42 just seemed to make sense.
01:01:44 So I would say,
01:01:47 this is also something that neuroscientists talk about,
01:01:50 to keep giving yourself questions
01:01:53 is a really good way to study.
01:01:56 Rather than looking through lots of material,
01:01:59 it’s always like giving yourself lots of tests
01:02:02 is a good way to actually deeply understand things
01:02:05 and know what you do and you don’t understand.
01:02:07 So would the questions be in the form of
01:02:11 the material you’re reviewing
01:02:12 is the answer to that question?
01:02:14 Or is it almost like beyond,
01:02:16 it’s the polygon thing they mentioned for a square.
01:02:18 Is it almost like, I wonder what is the bigger picture?
01:02:22 I was kind of asking like, how is this extended and so on?
01:02:26 Yeah, that would be great.
01:02:28 And it’s a similar,
01:02:30 I mean, a question I get asked a lot is about homework.
01:02:32 What is a good thing for kids to do for homework?
01:02:34 And one of the recommendations I give
01:02:36 is to not have kids just do lots of questions for homework,
01:02:42 but to actually ask them to reflect on what they’ve learned.
01:02:46 Like, what was the big idea you learned today?
01:02:49 Or what did you find difficult?
01:02:51 What did you struggle with?
01:02:53 What was something that was exciting?
01:02:58 Then kids go home and they have to kind of reflect
01:03:01 in a deeper way.
01:03:02 A lot of times, I don’t know if you had this experience
01:03:04 as a math student, lots of people do.
01:03:06 Kids are going through math questions,
01:03:08 they’re successful, they get them right,
01:03:10 but they don’t even really know what they’re about.
01:03:13 And a lot of kids go through many years of maths like that,
01:03:17 doing lots of questions,
01:03:18 but that really knowing what even the topic is
01:03:21 or what it’s about, what it’s important for.
01:03:24 So having students go back and think at the end of a day,
01:03:28 what was the big idea from this maths lesson?
01:03:31 Why is it important?
01:03:33 Where would I find that in real life?
01:03:36 Those are really good questions
01:03:37 for kids to be thinking about.
01:03:39 It’s probably for everybody to be thinking about.
01:03:42 I think most of us go through life
01:03:44 never asking the bigger question,
01:03:48 always those layers of why questions
01:03:51 that kids ask when they’re very young.
01:03:54 We need to keep doing that.
01:03:55 Like whatever the term is,
01:03:59 you call first principles thinking,
01:04:01 some people call it that,
01:04:03 which is like, why are we doing it this way?
01:04:08 So one nice thing is to do that
01:04:11 because there’s usually a good answer.
01:04:13 The reason we did it this way
01:04:15 is because it works for this reason.
01:04:17 But then if you want to do something totally novel,
01:04:20 you’ll say, well, we’ve been doing it this way
01:04:26 because of historical reasons,
01:04:28 but really this is not the best way to do it.
01:04:30 There might be other ways.
01:04:32 And that’s how invention happens.
01:04:34 And then you get, that’s really useful
01:04:36 in every aspect of life, like choosing your career,
01:04:40 choosing your, I don’t know, where you live,
01:04:44 who your romantic partner is, like everything.
01:04:47 Everything, yeah.
01:04:48 And I think it probably starts doing that in math class.
01:04:54 That would be good if we started doing that.
01:04:56 I mean, I wonder, I probably didn’t do very much of that
01:05:00 for most of my education, asking why,
01:05:03 except for later, much later in the subjects
01:05:08 on grad school when you’re doing research on them.
01:05:12 When you’re first tasked with doing something novel
01:05:15 using this or solving a problem
01:05:18 really outside the classroom,
01:05:19 they have to publish on it.
01:05:20 It’s the first time you think,
01:05:22 wait, why are these things interesting, useful?
01:05:27 Which are the things that are useful?
01:05:29 And yeah, I guess that would be nice
01:05:31 if we did that much earlier, the quest of invention.
01:05:35 Yeah, yeah, I mean, one of the sad pieces of research data
01:05:39 I think about is the questions kids ask in school goes down
01:05:45 like in a linear progression from, in the early years,
01:05:51 you can’t stop kids asking those questions,
01:05:53 but they learn not to ask the questions.
01:05:56 I think you told somewhere about an early memory
01:06:00 you had in your own education where you asked the question,
01:06:04 or maybe that was an example you gave,
01:06:06 but it was shut down.
01:06:07 Oh, yeah.
01:06:08 You’ve listened to something I said, yeah.
01:06:12 I don’t remember where it was, but it caught me.
01:06:15 Yeah, I remember it really vividly.
01:06:17 Or can you tell the memory?
01:06:19 Yeah, I was, it’s funny, I can remember.
01:06:21 It must’ve really impacted me in that moment
01:06:23 because you know how there’s lots of hours of school
01:06:26 you don’t remember at all, but anyway,
01:06:29 I can remember where I was sitting and everything.
01:06:30 I was in a high school maths class,
01:06:33 although they don’t call it that in England,
01:06:35 and the teacher said,
01:06:39 and it was like the first class of this teacher’s class,
01:06:41 and he said, ask if you have any questions.
01:06:44 So at one point I put my hand up and I said,
01:06:46 I have a question, and he said something like,
01:06:49 that’s your question?
01:06:53 And I was like, oh, okay,
01:06:55 I’m not asking any more questions in this class.
01:06:57 And that hit hard in a way where you didn’t wanna,
01:06:59 the lesson you learn from that is I’m not gonna ask.
01:07:02 Yeah, that was absolutely the last question I’m asking.
01:07:06 And that was, yeah, he was the chair of the maths department.
01:07:12 I remember that really well.
01:07:13 So maybe because of that experience,
01:07:18 one of the things we encourage when we teach kids
01:07:20 is asking questions, and we value it when they ask questions
01:07:23 and we put them up on walls and celebrate.
01:07:27 It’s funny because I wish there was a feedback signal
01:07:31 because he probably, to put a positive spin on it,
01:07:35 he probably didn’t realize the negative impact
01:07:37 he’s had in that moment, right?
01:07:38 If he only knew, see, this is probably
01:07:41 when you’re more mature in grad school.
01:07:44 I had an amazing professor named Ali Shakafande
01:07:46 in computer science, and he would get,
01:07:49 he would encourage questions,
01:07:51 but then he would tell everybody
01:07:52 how dumb their questions are.
01:07:54 But it was done, I guess if you show,
01:07:57 if you say it with love and respect behind it,
01:08:00 then it’s more like a friendly, humorous encouragement
01:08:03 for more questions.
01:08:05 Yeah, it’s an art, right, to do it, to write.
01:08:07 And then you have to time it right
01:08:10 because that kind of humor is probably better
01:08:13 for when you’re in grad school
01:08:15 versus when you’re in the early education.
01:08:18 Right.
01:08:18 Well, and I guess kids or young people get
01:08:22 whether somebody’s doing it to be funny or, you know,
01:08:27 I mean, this is why teaching is so hard.
01:08:31 Even your tone can be impactful.
01:08:35 It’s so sad because for that particular human,
01:08:39 the teacher, you just had a bad day,
01:08:42 and one statement can have a profound negative impact.
01:08:45 I know, sadly, that there’s a lot of maths teachers
01:08:50 who have that kind of approach,
01:08:51 and I think they’re suffering from the fact
01:08:56 that they think people are math people, not math people,
01:08:58 and that comes across in their teaching.
01:09:01 But on the flip side, one positive statement.
01:09:03 Yeah.
01:09:04 Keep them going.
01:09:05 That’s right.
01:09:05 That is the flip side of that.
01:09:07 And I myself had one teacher who was really amazing
01:09:11 for me in maths, and she kept me in the subject.
01:09:14 I probably wouldn’t have left it.
01:09:16 Who was she?
01:09:17 She was, her name was Mrs. Marshall.
01:09:20 And she was my A level maths teacher.
01:09:27 So I was in England.
01:09:30 You do lots of subjects till you’re 16,
01:09:32 and then you choose like three or four subjects.
01:09:35 So I had chosen maths, and you go to higher levels,
01:09:41 probably equivalent more to a master’s degree in the US
01:09:44 because you’re more specialized.
01:09:46 But anyways, she was my teacher,
01:09:48 and for the first time in my whole career in maths,
01:09:52 she would give us problems
01:09:54 and tell us to talk about them with each other.
01:09:58 And so here I was sitting there at like 17,
01:10:01 talking with friends about how to solve a math problem,
01:10:04 and that was it.
01:10:04 That was the change that she made,
01:10:07 but it was profound for me
01:10:10 because like those calculus students,
01:10:13 I started to hear other people’s ways of thinking
01:10:16 and seeing it, and we would talk together
01:10:17 and come up with solutions.
01:10:19 And I was like, that was it.
01:10:20 That changed maths for me.
01:10:23 It wasn’t some kind of personal interaction with her.
01:10:25 It was more like she was the catalyst
01:10:29 for that collaborative experience.
01:10:32 I mean, yeah, the many ways teachers can inspire kids.
01:10:34 I mean, sometimes it’s a personal message,
01:10:36 but it can be your teaching approach
01:10:39 that changes maths for kids.
01:10:42 You know, Cal Newport, he wrote a book called Deep Work,
01:10:48 and he’s a mathematician, a theoretical computer scientist,
01:10:51 and he talks about the kind of the focus required
01:10:55 to do that kind of work.
01:10:58 Is there something you can comment on?
01:11:01 You know, we live in a world full of distractions.
01:11:04 That seems like one of the elements that makes studying,
01:11:08 and especially the studying of subjects
01:11:10 that require thinking like maths does, difficult.
01:11:14 Is there something from a student perspective,
01:11:17 from a teacher perspective that encourages deep work
01:11:20 that you can comment on?
01:11:21 Yeah, I think giving kids really inspiring deep problems,
01:11:28 and we have some on our website,
01:11:31 is a really important experience for them.
01:11:35 Even if they only do it occasionally,
01:11:38 but it’s really important.
01:11:40 They actually realise, I do, I give a problem out often
01:11:44 when I’m working with teachers, and I say to them,
01:11:47 all right, I’m gonna check in with you after an hour.
01:11:49 And they’re like, an hour?
01:11:51 They think it’s shocking.
01:11:53 And then they work on this problem,
01:11:55 and after an hour, I say, okay, how are we doing?
01:11:57 They’re like, an hour’s gone by?
01:12:00 How is this possible?
01:12:02 And so everybody needs those like rich deep problems.
01:12:07 Most kids go through their whole maths experience
01:12:10 of however many years, never once working on a problem
01:12:14 in that kind of deep way.
01:12:16 So the undergrad class I teach at Stanford, we do that.
01:12:21 We work on these deep problems every session.
01:12:25 And the students come away going, okay,
01:12:27 I never wanna go back to that maths relationship I had
01:12:30 where it was just all about quick answers.
01:12:32 I just don’t wanna go back to that.
01:12:34 And so we can all, all teachers can incorporate
01:12:39 those problems in their classrooms.
01:12:41 Maybe they don’t do them every day,
01:12:42 but they at least give kids some experience
01:12:46 of being able to work slowly and deeply
01:12:49 and to go to deeper places and not be told
01:12:55 they’ve got five minutes to finish 20 questions.
01:12:58 Well, part of it is also just the exercise
01:13:02 of sitting there and maintaining focus
01:13:05 for prolonged periods of time.
01:13:07 That’s not often, I mean, that’s a skill.
01:13:11 It’s a skill that also could be discouraging.
01:13:16 Like if you don’t practice it,
01:13:18 just sitting down for 10 minutes straight
01:13:20 and maintaining deep focus could be exceptionally challenging.
01:13:23 Like if you’re really thinking about a problem
01:13:26 and I think it’s really important to realize
01:13:29 that that’s a skill that you can just like a muscle,
01:13:31 you can build, you can start with five minutes
01:13:33 and goes to 10 minutes to 30 and to an hour.
01:13:36 And to be successful, I think in certain subjects
01:13:39 like mathematics, you wanna be able to develop that skill.
01:13:42 Otherwise you’re not going to get
01:13:46 to the really rewarding experience
01:13:49 of solving these problems.
01:13:52 Definitely.
01:13:53 There was a survey done of kids in school
01:13:55 where they were asked, how long will you work
01:13:57 on a maths problem before you give up
01:13:59 and decide it’s not possible to solve it?
01:14:02 And the result on average across the kids was two minutes.
01:14:10 Yeah, that’s a bad sign, but that was a powerful sign
01:14:15 that they need to learn to not give up so quickly.
01:14:19 Yeah, we mentioned offline
01:14:22 because we’ve been talking so much about visualization,
01:14:25 Grant Sanderson, Three Blue One Brown.
01:14:29 So he’s inspired millions of people
01:14:32 with exactly the kind of way of thinking
01:14:35 that you’ve been talking about.
01:14:36 Yeah, I love his work.
01:14:38 Converting sort of mathematical concepts
01:14:42 into visual, like visually representing them,
01:14:48 exploring them in ways that help you illuminate
01:14:51 like the concepts.
01:14:53 What do you think is the role of that?
01:14:55 So he uses mostly programmatic visualization.
01:14:58 So it’s the thing I mentioned where there’s like animations
01:15:01 created by writing computer programs.
01:15:05 Like what do you think, how scalable is that approach?
01:15:08 But in general, what do you think about his approach?
01:15:10 I think it’s amazing.
01:15:11 I should work with him.
01:15:14 I can share some of our visuals
01:15:17 and he can make them in that amazing way.
01:15:20 So part of his storytelling,
01:15:23 part of it is creating the visuals
01:15:26 and then weaving a story with those visuals
01:15:29 that kind of builds, like there’s also,
01:15:32 I mean, there’s also drama in it.
01:15:34 You start with a small example
01:15:36 and then you kind of, all of a sudden there’s a surprise.
01:15:39 Yeah, yeah, yeah.
01:15:41 And it really, I mean, it makes you fall in love
01:15:43 with the concept.
01:15:45 He does talk about that.
01:15:48 His sense is like some of the stuff,
01:15:50 he doesn’t feel like he’s teaching like the core curriculum,
01:15:57 which is something, he sees himself
01:16:01 as an inspirational figure.
01:16:03 But because I think it’s too difficult
01:16:05 to kind of convert all of the curriculum into those elements.
01:16:09 And probably you don’t need to.
01:16:11 I mean, if people get to experience mathematical ideas
01:16:15 in the way that he shares them, that will change them.
01:16:20 And it will change the way they think.
01:16:21 And maybe they could go on
01:16:22 to take some other mathematical idea
01:16:25 and make it that beautiful.
01:16:26 Well, he does that.
01:16:28 He created a library called Manim and he open sourced it.
01:16:32 And that library is the, people should check it out.
01:16:36 It’s written in Python
01:16:39 and it uses some of those same elements.
01:16:41 Like it allows you to animate equations
01:16:44 and animate little shapes.
01:16:45 Like people that, you know,
01:16:47 he has a very distinct style in his videos
01:16:50 and what that resulted in,
01:16:51 even though from a software engineer perspective,
01:16:54 the code he released is not like super well documented
01:16:57 or perfect, but him releasing that,
01:17:01 now there’s all of these people educating it.
01:17:05 And the cool, to me personally,
01:17:06 the coolest thing is to see like people they’re not,
01:17:11 you know, don’t have like a million subscribers
01:17:13 or something is they have just a few views in the video,
01:17:17 but it just seems like the process of them
01:17:21 creating a video where they teach
01:17:24 is like transformative to them from a student perspective.
01:17:27 It’s the old Feynman thing,
01:17:28 the best way to learn is to teach.
01:17:30 And then him releasing that into the wild is,
01:17:33 it shows that that impact.
01:17:35 Yeah, absolutely.
01:17:37 I think just giving people that idea
01:17:39 that you can do that with maths and other subjects,
01:17:42 there’s bound to be people all around
01:17:45 who can create more, which is cool.
01:17:48 Yeah, definitely.
01:17:49 So I recommend that people do like JavaScript or Python.
01:17:52 You can build like visualizations of most concepts
01:17:56 in high school math.
01:17:57 You can do a lot of kinds of visualizations
01:18:00 and doing that yourself.
01:18:01 Plus, if you do that yourself, people will really love it.
01:18:04 People actually, people love visualizations of math.
01:18:09 Cause they, I mean, it’s something in us
01:18:11 that loves patterns, loves figuring out difficult things
01:18:14 and the patterns in there then are unexpected in some way.
01:18:19 Yeah.
01:18:19 Have you ever noticed that hotels
01:18:22 are always filled with patterns?
01:18:24 I was just noticing at the hotel I’m in now,
01:18:26 all of their carpets are pattern carpets
01:18:28 and then they have patterns on the walls.
01:18:30 Yeah.
01:18:31 So, yeah.
01:18:33 We humans love the symmetry and patterns,
01:18:36 the breaking of symmetry and patterns.
01:18:38 Yeah.
01:18:39 And then it’s funny that we don’t see mathematics
01:18:42 as somehow intricately connected to that, but it is, right?
01:18:45 I mean, that’s one of the perspectives
01:18:47 that I love students to take is to be a pattern seeker.
01:18:51 In everything.
01:18:51 In, yeah, certainly in all of maths.
01:18:53 I mean, you can think of all of maths
01:18:55 as a kind of subject of patterns
01:18:57 and not just visual patterns, but, you know,
01:19:01 when you think about multiplying by five
01:19:05 and the fact you can, you know,
01:19:08 if you’re multiplying 18 times five,
01:19:11 you can instead think of nine times 10.
01:19:14 That’s a pattern that always works in mathematics.
01:19:17 You can halve a number and double a number.
01:19:19 And so, yeah, I just think there are patterns everywhere.
01:19:21 And if kids are thinking their role is to see patterns
01:19:24 and find patterns, it’s really exciting.
01:19:28 What do you think about like MIT OpenCourseWare
01:19:31 and the release of lectures by universities?
01:19:36 I think it’s good.
01:19:37 I think it’s good.
01:19:38 I think that is what started the MOOC I did
01:19:43 was using that platform.
01:19:46 So you ultimately think like the Udacity models
01:19:48 is a little bit more effective
01:19:50 than just a plain two hour lecture.
01:19:52 I think there’s definitely,
01:19:53 you can bring in good pedagogy into online learning.
01:19:57 And I think the idea of putting things online
01:19:59 so that people all over the world can access them is great.
01:20:02 I don’t think the initial excitement around MOOCs
01:20:06 sort of democratizing education
01:20:08 and make it more equal came about
01:20:11 because they found that the people taking MOOCs
01:20:14 tended to be the more privileged people.
01:20:17 So that was, I think there’s still something to be found
01:20:20 in that there’s still more to be done
01:20:22 to help that online learning reach those principles.
01:20:27 But definitely, I think it’s a good invention.
01:20:31 And I have an online class that’s for kids,
01:20:35 that’s a little free class that gives them.
01:20:36 What’s the topic?
01:20:37 It’s called How to Learn Maths.
01:20:39 How to Learn Maths.
01:20:41 It shows maths as this visual creative subject
01:20:44 and it shares mindset and some brain science
01:20:47 and kids who take it do better in maths class.
01:20:51 We’ve studied it with like randomized controlled trials
01:20:54 and given it to middle school kids
01:20:57 and other middle school kids who don’t take it
01:21:00 but are taught by the same teachers.
01:21:01 So their teachers are the same.
01:21:02 And the kids who take the online class
01:21:04 end up 68% more engaged in their maths class
01:21:08 and do better at the end of the year.
01:21:10 So that’s a little six session, 15 minute class
01:21:15 and it changes kids maths relationships.
01:21:18 So it is true that we can do that with some words
01:21:24 that aren’t, it’s not a huge change to the education system.
01:21:30 Do you have advice for young people?
01:21:33 We’ve been talking about mathematics quite a bit
01:21:35 but in terms of their journey through education,
01:21:39 through their career choices, through life,
01:21:41 maybe middle school, high school, undergrad students,
01:21:46 of how to live a life they can be proud of?
01:21:50 I think if I were to give advice to people,
01:21:54 especially young people, my advice would be to always,
01:21:59 it sounds really corny,
01:22:00 but always believe in yourself
01:22:03 and know that you can achieve
01:22:05 because although that sounds like obvious,
01:22:07 of course we want kids to know that they can achieve things.
01:22:10 I know that millions of kids who are in the school system
01:22:13 have been given the message, they cannot do things.
01:22:17 And adults too, they have the idea,
01:22:20 oh, I did okay in this, I went into this job
01:22:23 because those other things I could never have done okay in.
01:22:26 So actually when they hear,
01:22:28 hey, maybe you could do those other things.
01:22:31 Even adults think, maybe I can.
01:22:34 And they go back and they encounter this knowledge
01:22:38 and they relearn things and they change careers
01:22:40 and amazing things happen.
01:22:42 So for me, I think that message is really important.
01:22:45 You can learn anything.
01:22:47 Scientists try and find a limit.
01:22:50 They’re always trying to find a limit,
01:22:51 like how much can you really learn?
01:22:52 What’s the limit to how much you can learn?
01:22:54 And they always come away not being able to find it.
01:22:56 People can just go further and further and further.
01:22:59 And that is true of people born with brain,
01:23:04 areas of their brain that aren’t functioning well
01:23:06 that have what we call special needs.
01:23:08 Some of those people also go on to develop
01:23:10 and do amazing things.
01:23:12 So I think that really experiencing that,
01:23:17 knowing that feeling, not just saying it,
01:23:19 but knowing it deeply, you can learn anything
01:23:24 is something I wish all people would have.
01:23:28 Actually also applies when you’ve achieved
01:23:30 some level of success too.
01:23:32 What I find, like in my life with people that love me,
01:23:36 when you achieve success,
01:23:37 they keep celebrating your success
01:23:40 and they want you to keep doing the thing
01:23:42 that you were successful at,
01:23:44 as opposed to believing in that you can do something else,
01:23:49 something big, whatever your heart says to do.
01:23:51 And one of the things that I realized the value of this,
01:23:57 quite recently, which is sad to say,
01:24:00 is how important it is to seek out,
01:24:03 when you’re younger, to seek out mentors,
01:24:04 to seek out the people,
01:24:07 surround yourself with people that will believe in you.
01:24:10 It’s like a little bit is on you.
01:24:14 It’s like, you don’t get that sometimes
01:24:18 if you go to grad school,
01:24:20 you think you kind of land on a mentor,
01:24:22 maybe you pick a mentor based on the topic
01:24:24 they’re interested in.
01:24:25 But the reality is the people you surround yourself with,
01:24:28 they’re going to define your life trajectory.
01:24:32 So select people that believe in you.
01:24:35 And get away from people who don’t believe in you.
01:24:38 Sometimes parents can be that, they can love you deeply,
01:24:41 but they set, it’s the math thing we mentioned,
01:24:46 they might set certain constraints on the beliefs
01:24:50 that you have.
01:24:50 And so in that, if you’re interested in mathematics,
01:24:55 and your parents are not that interested in it,
01:24:56 don’t listen to your parents on that one dimension.
01:24:59 Exactly.
01:25:00 Yeah, and if people tell you you can’t do things,
01:25:02 you have to hear from other people who believe in you.
01:25:06 I think you’re absolutely right about that.
01:25:09 So sad the number of people who’ve had
01:25:11 those negative messages from parents.
01:25:14 In my Limitless Mind book, I interviewed quite a few people
01:25:16 who’d been told they couldn’t do math,
01:25:18 sometimes by parents, sometimes by teachers.
01:25:21 And fortunately, they had got other ideas
01:25:24 at some point in their life,
01:25:26 and realized there was this whole world
01:25:28 of mathematical thinking that was open to them.
01:25:32 So it’s really important that people do connect
01:25:37 with people who believe in them,
01:25:39 however hard that might be to find those people.
01:25:41 What do you hope the education system,
01:25:44 education in general, looks like 10, 20, 50,
01:25:48 100 years from now?
01:25:49 Are you optimistic about this future?
01:25:51 Yeah, I definitely have hope.
01:25:53 There is, change can happen in the education system.
01:25:56 In recent years, it’s been microscopically slow.
01:26:03 But I do actually see change happening.
01:26:07 We were talking earlier that data science is now
01:26:10 a course you can take in high school instead of algebra two.
01:26:14 And that’s pretty amazing because that content
01:26:17 was set out in 1892 and hasn’t changed since then.
01:26:22 And so now we’re actually seeing a change
01:26:25 in the content of high school.
01:26:26 So I’m amazed that that’s happening
01:26:29 and very happy it’s happening.
01:26:31 So change is very slow in education usually,
01:26:33 but when you look ahead and think about all that we know,
01:26:38 and all that we can offer kids in terms of technology,
01:26:42 you’ve got to think that 100 years from now,
01:26:46 education will be totally different to the way it is now.
01:26:50 Maybe we won’t have subject boundaries anymore
01:26:54 because those don’t really make much sense.
01:26:57 And it’s interesting to think how certain tools
01:27:00 like programming, maybe they’ll be deeply integrated
01:27:03 in everything we do.
01:27:04 Yeah, you would think that all kids are growing up
01:27:07 learning to program and create.
01:27:11 So I just think, I mean, the system of schooling
01:27:14 we have now, people call it a factory model.
01:27:17 It’s not designed to inspire creativity.
01:27:21 And I feel like that will also change.
01:27:25 People might look back on these days
01:27:27 and think they were hilarious,
01:27:28 but maybe we’ll in the future,
01:27:32 kids will be doing their own programming
01:27:33 and they’ll be able to learn things
01:27:35 and find out things and create things
01:27:37 even as they’re learning.
01:27:37 And maybe the individual subject boundaries will go.
01:27:44 Data science itself coming into the education system
01:27:47 kind of illustrates that because people realize
01:27:50 it doesn’t really fit inside any of the subjects.
01:27:54 So what do we do with it?
01:27:56 Where does it go and who teaches it?
01:27:59 So it’s already raising those kind of questions
01:28:03 and questioning how we have these different subject boundaries.
01:28:07 So you’ve seen data science be integrated
01:28:09 into the curriculum?
01:28:10 Yes, it’s happening across the United States as we speak.
01:28:14 I wonder how they got initiated.
01:28:15 Like how does change happen in the education system?
01:28:18 Is it just a few revolutionary leaders?
01:28:20 It does, I think so.
01:28:22 I think so.
01:28:23 It’s been an interesting journey
01:28:24 seeing data science take off actually.
01:28:26 There was a course that was developed in 2014
01:28:32 by some people who thought data science
01:28:34 was a good idea for high schoolers.
01:28:36 And then after some kids took the course
01:28:39 and nothing bad happened to them,
01:28:41 they went to college and people started to accept it more.
01:28:46 And then this was a big piece of the change in California.
01:28:49 The UC system communicated.
01:28:52 They sent out an email last year,
01:28:53 the 50,000 high schools saying, we now accept data science.
01:28:57 Kids can take it instead of algebra two.
01:28:59 That’s a perfectly legitimate college pathway.
01:29:01 So that was like a big green light for a lot of schools
01:29:05 who were like wondering about whether they could teach it.
01:29:08 So I think it happens in small spaces and expands.
01:29:12 It goes viral.
01:29:13 Yeah.
01:29:14 In this modern age.
01:29:15 Then it goes viral.
01:29:16 California’s ahead, I think in creating courses
01:29:21 and having kids go through it, but it’s suddenly,
01:29:25 when I last looked, there were 12 states
01:29:27 that were allowing data science as a high school course.
01:29:29 And I think by next year, that will have doubled or more.
01:29:35 So change is happening.
01:29:38 Joe, as I said, I think mathematics
01:29:41 is truly a beautiful subject.
01:29:45 And you having an impact on millions of people
01:29:48 and you having an impact on millions of people’s lives
01:29:52 by educating them, by inspiring teachers to educate
01:29:56 in the ways that you’ve talked about
01:29:57 in multidimensional ways, in visual ways,
01:30:01 I think is incredible.
01:30:02 So you’re spreading beauty into the world.
01:30:06 So I really, really appreciate
01:30:07 that you spend your valuable time with me today.
01:30:09 Thank you for talking.
01:30:10 Thank you.
01:30:10 It was really good to talk to you.
01:30:12 Thanks for listening to this conversation with Joe Boller.
01:30:15 To support this podcast,
01:30:16 please check out our sponsors in the description.
01:30:19 And now let me leave you with some words
01:30:21 from Albert Einstein.
01:30:23 Pure mathematics is the poetry of logical ideas.
01:30:27 Thanks for listening and hope to see you next time.