Jo Boaler is a professor of mathematics education at Stanford and the co-founder of youcubed. Please support this podcast by checking out our sponsors:
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OUTLINE:
0:00 - Introduction
0:23 - What is beautiful about mathematics?
9:12 - How difficult should math really be?
17:31 - Students giving up on math
28:52 - Improving math education in schools
38:49 - Inspiring mathematical creativity
56:35 - youcubed
1:00:55 - Best methods for studying math
1:21:29 - Advice for young people

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The following is a conversation with Jo Bowler, a mathematics educator at Stanford and co-founder of ucubed.org that seeks to inspire young minds with the beauty of mathematics.

To support this podcast, please check out our sponsors in the description.

This is the Lex Friedman Podcast, and here is my conversation with Jo Bowler.

What to you is beautiful about mathematics?

I love a mathematics that some people don't even think of as mathematics, which is beautiful, creative mathematics, where we look at maths in different ways, we visualize it, we think about different solutions to problems.

A lot of people think of maths as you have 1 method and 1 answer, and what I love about maths is the multiple different ways you can see things.

Different methods, different ways of seeing, in some cases, different solutions.

So that is what is beautiful to me about mathematics, that you can see and solve it in many different ways.

And also the sad part that many people think that maths is just 1 answer and 1 method.

So to you, the beauty emerges when you have a problem with a solution and you start adding other solutions, simpler solutions, weirder solutions, more interesting, some that are visual, some of their algebraic, geometry, all that kind of stuff.

Yeah, I mean, I always say that you can take any maths area and make it visual.

And we say to teachers, give us your most dry, boring maths, and we'll make it a visual, interesting, creative problem.

And it turns out you can do that with any area of maths.

And I think we've given, it's been a great disservice to kids and others that it's always been numbers, lots and lots of numbers.

Numbers can be great, but you can think about maths in other ways besides numbers.

Do you find that most people are better visual learners, or is this just something that's complimentary?

What's the kind of the full spectrum of students in the way they like to explore math, would you say?

There's definitely people who come into the classes I do who are more interested in visual thinking and like visual approaches.

But it turns out what the neuroscience is telling us is that when we think about maths, there are 2 visual pathways in the brain and we should all be thinking about it visually.

Some approaches have been to say, well, you're a visual learner, so we'll give you visuals, and you're not a visual learner.

But actually, if you think you're not a visual learner, it's probably more important that you have a visual approach so you can develop that part of your brain.

So you were saying that there's some kind of interconnected aspect to it, so the visual connects with the non-visual.

Yeah, so this is what the neuroscience has shown us, that when you work on a math problem, there are 5 different brain pathways, and that the most high achieving people in the world are people who have more connections between these pathways.

So if you see a maths problem with numbers, but you also see it visually, that will cause a connection to happen in your brain between these pathways.

And if you maybe write about it with words, that would cause another connection, or maybe you build it with something physical, that would cause a different connection.

And what we want for kids is, we call it a multi-dimensional experience of math.

Seeing it in different ways, experiencing it in different ways.

That will cause that great connected brain.

You know, there's these stories of physicists doing the same.

I find physicists are often better at building that part of their brain of using visualization for intuition building, because you ultimately want to understand the deepest secret underneath this problem.

And for that, you have to intuit your way there.

And you mentioned offline that 1 of the ways you might approach a problem is to try to tell a story about it.

And some of it is like legend, but I'm sure it's not always, is you have Einstein thinking about a train, and the speed of light, and that kind of intuition is useful.

You start to imagine a physical world, like how does this idea manifest itself in the physical world, and then start playing in your mind with that physical world, and think is this going to be true, is this going to be true?

Einstein is well known for thinking visually.

And people talk about how he really didn't want to go anywhere with problems without thinking about them visually.

But the other thing you mentioned that sparked something for me is thinking with intuition, like having intuition about math problems.

That's another thing that's often absent in math class, the idea that you might think about a problem and use your intuition, but so important.

And when mathematicians are interviewed, they will very frequently talk about the role of intuition in solving problems, but not commonly acknowledged or brought into education.

Like, if you task yourself with building an intuition about a problem, that's where you start to pull in like what is the pattern I'm seeing?

In order to understand the pattern, you might want to then start utilizing visualization.

But ultimately that's all in service of like solving the puzzle, like cracking it open to get the simple explanation of why things are the way they are, as opposed to, like you said, having a particular algorithm that you can then execute to solve the problem.

I mean, I love to value what's hard in maths instead of being afraid of it.

We know that when you struggle, that's actually a really good time for your brain.

You want to be struggling when you're thinking about things.

So if it's hard to think intuitively about something, that's probably a really good time for your brain.

I used to work with somebody called Sebastian Thrun, who is a great mathematician, you might think of him, an AI person.

I remember in 1 interview I did with him, he talked about how they'd built robots, I think for the Smithsonian, and how they were having this trouble with them picking up white noise.

And he said they had to solve it, they had to work out what's going on, and how he intuitively worked out what the problem was.

But then it took him 3 weeks to show it mathematically.

I thought that was really interesting that how you can have this intuition and know something works.

It's kind of different from going through that long mathematical process of proving it, but so important.

Yeah, I think probably our brains are evolved as like intuition machines.

And the math of like showing it like formally is probably an extra thing that we're not designed for.

You see that with Feynman and his, I mean, it just all of these physicists, definitely you see starting with intuition, sometimes starting with an experiment, and then the experiment inspires intuition.

But you can think of an experiment as a kind of visualization.

Just like, let's take whatever the heck we're looking at and draw it and draw like the pattern as it evolves, as the thing grows for N equals 1, for N equals 2, N equals 3, you start to play with it.

And then in the modern day, which I loved doing, is you can write a program that then visualizes it for you.

And then you can start exploring it programmatically.

And then you can do so interactively too.

I tend to not like interactive because it takes way too much work because you have to click and move and stuff.

I love to interact through writing programs, but that's my particular brain, software engineer.

So like you can do all these kinds of visualizations And then there's the tools of visualization, like color, all those kinds of things.

That you're absolutely right, they're actually not taught very much.

Like when you represent something mathematically, you can show color to show the growth and kind of code that.

So if I have an algebraic expression for a pattern, maybe I show the X with a certain color, but also right in that color so you can see the relationship.

And yeah, particularly in our work with elementary teachers, many of them come to our workshops and they're literally in tears when they see things making sense visually.

Because they've spent their whole lives not realizing you can really understand things with these visuals.

You say that there's something about, there's something valuable to learning when the thing that you're doing is challenging, is difficult.

So a lot of people say, you know, math is hard, or math is too hard, or too hard for me.

Do you think math should be easy, or should it be hard?

I think it's great when things are challenging, but there's something that's really key to being able to deal with challenging maths, and that is knowing that you can do it.

And I think the problem in education is a lot of people have got this idea that you're either born with a math brain or you're not.

So when they start to struggle, they think, oh, I don't have that math brain.

And then they will literally sort of switch off in their brain and things will go downhill from that point.

So struggle becomes a lot easier and you're able to struggle if you don't have that idea.

You have to go through this struggle to get there, but you're able to do that.

And so we're hampered in being able to struggle with these ideas we've been given about what we can do.

So there's kind of, I don't know what the right term is, but some people are struggle with learning in different ways.

Like their brain is constructed in different ways.

And how much should, as educators, should we make room for that?

So how do you know the difference between this is hard and I don't like doing hard things versus my brain is wired in a way where I need to learn in very different ways, I can't learn it this way.

So this is why being a teacher is so hard.

And people really don't appreciate how difficult teaching is when you're faced with, I don't know, 30 students who think in different ways.

But this is also why I believe it's so important to have this multi-dimensional approach to maths.

We've really offered it in 1 way, which is here's some numbers and a method, you follow me, do what I just did, and then reproduce it.

And so there are some kids who like doing that and they do well.

And a lot of kids who don't like doing it and don't do well.

But when you open up maths and you give, you let kids experience it in different ways, maybe visually, with numbers, with words, what happens is kids, there are many more kids who can access it.

So those different brain wiring you're talking about, where some people are just more able to do something in a particular way, That's why we want to, that's 1 of the reasons we want to open it up, so that there are different ways of accessing it.

So I grew up in the Soviet Union and fell in love with math early.

I was forced into math early and fell in love through force.

That's good, well, good that you fell in love and learned about

But there's something we talked about a little bit is there's such a value for excellence.

It's competitive and it's also everybody kind of looks up, the definition of success is being in a particular class is being really good at it.

And it's not improving, it's like being really good.

I mean, we are much more like that with sports, for example.

We're not, it's like it's understood, you're going to star on the basketball team, if you're gonna start on the basketball team, if you're going to be better than the other guys, the other girls on the team.

So that coupled with the belief, this could be partially a communist belief, I don't know, but the belief that everybody is capable of being great.

But if you're not great, that's your fault.

And I remember I had a sense that, probably delusional, but I could win a Nobel Prize.

But I thought, like my dad early on told me just offhand and it always stuck with me that if you can figure out how to build a time machine, how to travel back in time, it will probably give you a Nobel Prize.

And I remember early in my life thinking, I'm going to invent the time machine.

And like the tools of mathematics were in service of that dream of winning the Nobel Prize.

And it's silly, I didn't really think in those concrete terms, but I just thought I could be great, that feeling.

And then when you struggle, the belief that you could be great is like, struggle is good.

And so The other thing about the Soviet system that might love to hear your comments about is just the sheer like hours of math.

number of courses you're talking about, a lot of geometry, a lot more geometry.

I think in the American system, you take maybe 1 year of geometry.

In high school, first of all, geometry is beautiful.

And then you get to reason through proofs and stuff like that.

In Russia, I remember just being nailed over and over with geometry, it was just nonstop.

And then of course there's different perspectives on calculus and just the whole, the sense was that math is like fundamental to the development of the human mind.

So math, but also science and literature, by the way, was also hit very hard.

Like we read a lot of serious adult stuff.

They challenge young adults with good literature, but they don't challenge adults very much with math.

So those 2 things, valuing excellence and just a lot of math in the curriculum.

Do you think, do you find that interesting?

Because it seems to have been successful.

And there is a lot of success of people coming through the Soviet system.

I think something that's very different to the US and other countries in the world is this idea that excellence is important and you can get there if you work hard.

In the US, there's an idea that excellence is important, but then kids are given the idea in many ways that you can either do it or you're 1 of the people who can't.

So many students in the school system think they're 1 of the kids who can't.

So there's no point in trying hard because you're never going to get there.

So if you can switch that idea, it would be huge.

And it seems from what you've said that in the Soviet Union, that idea is really different.

Now, the downside of that idea that anybody can get there if you work hard is that thought that if you're not getting there, it's your fault.

I would say that anybody can get there, But they need to work hard and they also need good teaching.

Because there are some people who really can't get there because they're not given access to that good teaching.

As to doing lots of maths, If maths was interesting and open and creative and multidimensional, I would be all for it.

We actually run summer camps at Stanford where we invite kids in and we give them this maths that I love.

In our camp classrooms, they were 3 hours long.

And when we were planning, the teachers were like, 3 hours, are we gonna be able to keep the kids excited for 3 hours?

Turned out, they didn't want to go to break or lunch.

They'd be so into these mathematical patterns.

So yeah, if maths was more like that, then I think having more of it would be a really good thing.

Is there, could you comment on what age is like the most important when people quit math or give up on themselves or on math in general?

And perhaps that age or something earlier is really an important moment for them to discover, to be inspired to discover the magic of math.

I think a lot of kids start to give up on themselves and maths around from about fifth grade.

And then those middle school years are really important.

And fifth grade can be pivotal for kids just because they're allowed to explore and think in good ways in the early grades of elementary school, but fifth grade teachers are often like, okay, we're going to prepare you now for middle school, and we're going to give you grades and lots of tests.

And that's when kids start to feel really badly about themselves.

And so middle school years, our camps are middle school students.

We think of those years as really pivotal.

Many kids in those years are deciding, yes, I'm gonna keep going with STEM subjects, or no, I'm not, that this isn't for me.

So, I mean, all years are important, and in all years, you can kind of switch kids and get them on a different pathway, but I think those middle school years are really important.

So what's the role of the teacher in this?

So 1 is the explanation of the subject, but do you think teachers should almost do like one-on-one, you know, little Johnny, I believe in you kind of thing, that energy of like.

There's a study that was done, it was actually done in high school English classrooms, where all kids wrote an essay for their teacher.

Half of the kids got feedback from their teacher, diagnostic feedback, which is great.

But for half of the kids, it said an extra sentence at the bottom that the researchers had put on.

And the kids who read that extra sentence did significantly better in English a whole year later.

The sentence said, I'm giving you this feedback because I believe in you.

And the kids who read that did better a year later.

So when I share this with teachers, I say, you know, I'm not suggesting you put on the bottom of all kids' work.

I'm giving this feedback because I believe in you.

1 of the teachers said to me, we don't put it on a stamp?

And kids are sitting in classrooms all the time thinking, what does my teacher think of me?

So it turns out it is really important to be saying to kids, I know you can do this.

And those messages are not given enough by teachers.

Yeah, it's like, I just say it, you have to believe it.

I sometimes because like, it's such a funny dance, because I'm such a perfectionist, I'm extremely self-critical.

And I have 1 of the students come up to me and it's clear to me that they're not even close to good.

And it's tempting for me to be like, to sort of give up on them mentally.

But the reality is like, if you look at many great people throughout history, they sucked

And some of the greatest took non-linear paths to where they sucked for long into later life.

And so always kind of believing that this person can be great.

You have to communicate that, plus the fact that they have to work hard.

Silicon Valley, where I live, is filled with people who are dropouts at school or who had special needs, who didn't succeed.

It's very interesting that have gone on to do amazing work in creative ways.

I mean, I do think our school system is set up to value good memorizers who can reproduce what a teacher is showing them and push away those creative deep thinkers, often slower thinkers.

They think slowly and deeply, and they often get the idea early on that they can't be good at maths or other subjects.

So Yeah, I think many of those people are the ones who go on and do amazing things.

I know many mathematicians like this, but he talks a lot about having a non-standard way of learning.

I mean, a lot of great mathematicians, a lot of great physicists are like that.

And he felt like he became quickly, he got his PhD at Harvard, became quickly an outcast of the system.

Like the education, especially early education system didn't help him.

Is there ways for an education system to support people like that?

Is it this kind of multi-dimensional learning that you're mentioning?

I mean, I think Our education system still uses an approach that was in classrooms hundreds of years ago.

The textbooks have a lot to answer for in producing this very uninspiring mathematics.

But yeah, if you open up the subject and have people see and solve it in different ways and value those different ways.

Somebody I appreciated a lot is a mathematician called Mary Mizzikani.

First woman in the world to win the Fields Medal in mathematics.

And she talked about how her daughter thought she was an artist because she was always visualizing.

And she asked me to chair the PhD defense for 1 of her students, and I went to the defense in the math department, and it was so interesting because this young woman spent like 2 hours sharing her work, all of it was visual.

In fact, I don't think I saw any numbers at all.

And I remember that day thinking, wow, I could have brought a 13-year-old into this PhD defense, they would not recognize this as maths.

But when Maryam Mizrakhani won the Fields Medal, all these other mathematicians were saying that her work had connected all these previously unconnected areas of maths.

But she also shared that when she was in school, when she was about 13, she was told that she couldn't do maths.

So I love that, you know, to be told you can't be good at maths and then go on and win the Fields Medal is cool.

I've been told by a lot of people in my life that I can't do something.

But all it takes, that's why people talk about the 1 teacher that changed everything.

You as a single person, given the education system, given the incentives, you have the power to truly change lives.

And like 20 years from now, I feel as medalist will walk up to you and say thank you.

And I share that with teachers that even in this broken system of what they have to do for districts and textbooks, a single teacher can change kids' math relationship or other subjects and forever.

What's the role of the parents in this picture?

1 study found that the amount of maths anxiety parents had predicted their child's achievement in school, But only if they helped with homework.

Yeah, there are some interesting implications for this.

If you have maths anxiety and you're helping your kids with homework, you're probably communicating things like, I was terrible at this at school.

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