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Speaker 1

00:00

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?

Speaker 2

00:27

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.

Speaker 2

00:59

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.

Speaker 3

01:13

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.

Speaker 2

01:30

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.

Speaker 2

01:46

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.

Speaker 3

02:00

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?

Speaker 2

02:13

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.

Speaker 2

02:40

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.

Speaker 3

02:51

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

Speaker 2

02:56

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.

Speaker 2

03:28

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.

Speaker 3

03:40

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.

Speaker 3

03:59

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?

Speaker 2

04:31

KF1 Right. Einstein is well known for thinking visually. And people talk about how he really didn't want to go anywhere with problems without thinking about them visually.

Speaker 2

04:43

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.

Speaker 3

05:13

Yeah, I mean, that's what it is. 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.

Speaker 3

05:31

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. Yeah, but it's hard, it's hard. Like reasoning is really hard.

Speaker 2

05:53

Yeah, it's hard. I mean, I love to value what's hard in maths instead of being afraid of it. We know that when you struggle, that's actually a really good time for your brain.

Speaker 2

06:04

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.

Speaker 2

06:32

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.

Speaker 3

06:58

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.

Speaker 3

07:27

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.

Speaker 3

07:56

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.

Speaker 3

08:22

That you're absolutely right, they're actually not taught very much. Like the art of visualization.

Speaker 2

08:28

Not taught. And we love as well color coding. Like when you represent something mathematically, you can show color to show the growth and kind of code that.

Speaker 2

08:41

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. Very cool. 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.

Speaker 2

09:10

It's quite powerful.

Speaker 3

09:12

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?

Speaker 2

09:33

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.

Speaker 2

09:58

And then they will literally sort of switch off in their brain and things will go downhill from that point. So struggle becomes a lot easier and you're able to struggle if you don't have that idea. But you know that you can do it. You have to go through this struggle to get there, but you're able to do that.

Speaker 2

10:19

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

Speaker 3

10:26

Yeah. 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?

Speaker 3

10:49

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

Speaker 2

11:04

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.

Speaker 2

11:21

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.

Speaker 2

11:52

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. And then that's not really a problem.

Speaker 3

12:11

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.

Speaker 2

12:22

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

Speaker 3

12:25

the force. 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.

Speaker 3

12:47

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.

Speaker 3

13:19

But if you're not great, that's your fault. And you need to work harder. And I remember I had a sense that, probably delusional, but I could win a Nobel Prize. I don't even know what that entails.

Speaker 3

13:33

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.

Speaker 3

14:06

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

Speaker 2

14:13

That pushes you on, yeah.

Speaker 3

14:14

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.

Speaker 2

14:23

Like the

Speaker 3

14:24

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.

Speaker 2

14:33

In high school, yeah.

Speaker 3

14:33

In high school, first of all, geometry is beautiful. It's visual. And then you get to reason through proofs and stuff like that.

Speaker 3

14:40

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.

Speaker 3

15:08

America does that a little bit too. 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?

Speaker 3

15:26

Because it seems to have been successful.

Speaker 2

15:28

Yeah, I think that's very interesting. 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.

Speaker 2

15:45

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.

Speaker 2

16:09

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

Speaker 2

16:35

Because there are some people who really can't get there because they're not given access to that good teaching. But that would be huge, that change. 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.

Speaker 2

17:01

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.

Speaker 2

17:19

We couldn't stop them, it was amazing. So yeah, if maths was more like that, then I think having more of it would be a really good thing.

Speaker 3

17:29

So what age are you talking about? 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.

Speaker 2

17:51

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.

Speaker 2

18:18

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.

Speaker 2

18:38

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.

Speaker 3

18:48

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.

Speaker 2

19:03

Turns out it's really important. 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. And this was done as an experiment.

Speaker 2

19:16

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 only change was this 1 sentence.

Speaker 3

19:36

What did the sentence say?

Speaker 2

19:37

So what did the sentence say? The sentence said, I'm giving you this feedback because I believe in you. And the kids who read that did better a year later.

Speaker 3

19:48

Yeah.

Speaker 2

19:50

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?

Speaker 2

20:00

I said, no, don't put it on a stamp. But your words are really important. And kids are sitting in classrooms all the time thinking, what does my teacher think of me? Does my teacher think I can do this?

Speaker 2

20:17

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.

Speaker 3

20:27

I really believe it.

Speaker 2

20:28

And believe it.

Speaker 3

20:29

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.

Speaker 3

20:49

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

Speaker 2

21:00

at some point.

Speaker 3

21:01

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.

Speaker 3

21:17

That's it, yeah.

Speaker 2

21:19

Yeah, and you're right. Silicon Valley, where I live, is filled with people who are dropouts at school or who had special needs, who didn't succeed. It's very interesting that have gone on to do amazing work in creative ways.

Speaker 2

21:34

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.

Speaker 3

22:05

So there's a guy named Eric Weinstein. 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.

Speaker 3

22:22

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?

Speaker 2

22:43

Absolutely, absolutely. I mean, I think Our education system still uses an approach that was in classrooms hundreds of years ago. The textbooks have a lot to answer for in producing this very uninspiring mathematics.

Speaker 2

22:58

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. I don't know if you've heard of her. She won the Fields Medal.

Speaker 2

23:12

She was from Iran. First woman in the world to win the Fields Medal in mathematics. She died when she was 40. She was at Stanford.

Speaker 2

23:21

But her work was entirely visual. 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.

Speaker 2

23:51

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. She was told that by her teacher.

Speaker 3

24:22

This is Iran? And she grew up there?

Speaker 2

24:24

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

Speaker 3

24:33

I've been told by a lot of people in my life that I can't do something. I'm very definitely non-standard. But all it takes, that's why people talk about the 1 teacher that changed everything.

Speaker 3

24:46

All it takes is 1 teacher. That's the power of that. So that should be inspiring to teachers.

Speaker 2

24:55

I think it is.

Speaker 3

24:57

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.

Speaker 2

25:07

You did that for me. Yeah, absolutely. 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.

Speaker 3

25:26

What's the role of the parents in this picture? Let's go to another difficult subject.

Speaker 2

25:31

Yeah, that is a difficult subject. 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. So.

Speaker 3

25:54

Oh, that's so funny.

Speaker 2

25:55

Yeah, there are some interesting implications for this. I mean, you can see how it works. 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.

Speaker 2

26:09

And that's how it gets passed on to kids. So 1 implication is if You have a really bad relationship with maths, you hate maths, you have maths anxiety, just don't do maths so well with your kids. But we have, on our website, we have a little sheet for parents of ways to interact around maths with your kids.

Speaker 3

26:33

That's ucubed.org? That's ucubed.org, yes.

Speaker 2

26:39

So 1 of the things I say to parents when I give parent presentations is, even if you hate maths, you need to just fake it with your kids. You should be always endlessly optimistic and happy about doing maths. And...

Speaker 3

26:53

I'm always curious about this, so I hope to have kids 1 day, I don't have kids currently. Are parents okay with sucking at math and then trying to get their kid to be better than them essentially? Like, is that difficult thing for a lot of parents?

Speaker 3

27:09

It is difficult. To have like, it's almost like an ego thing. Like I never got good at this and I probably should have. And yeah, I mean, to me that you want to celebrate that, but I know a lot of people struggle with that, like coaches in sports, to make an athlete become better than them, it can be hard on the ego.

Speaker 3

27:31

Yeah. So do you experience the same with parents?

Speaker 2

27:35

I think, I mean I haven't experienced parents worrying that their kids will be better than them. I have experienced, I have experienced parents just having a really bad relationship with maths and not wanting to help, not knowing how to help, saying things. Like another study showed that when mothers say to their daughters, I was bad at maths in school, their daughter's achievement goes down.

Speaker 2

28:02

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

Speaker 2

28:31

There's 1 method that you memorize and then you reproduce it. So until people have really had an experience of what I think of as the other maths, where until they've really seen that it's a really different subject, It's hard for them to be able to shift their kids to see it differently.

Speaker 3

28:52

Is there for a teacher, if we're to like systematize it, is there something teachers can do to do this more effectively? So you mentioned the textbook. So what are the additional things you can add on top of this whole old school, traditional way of teaching that can improve the process?

Speaker 2

29:14

So I do think there's a way of teaching maths that changes everything for kids and teachers. I'm 1 of 5 writers of a new framework for the state of California, a new maths framework that's coming out next year. We are recommending through this maths framework that people teach in this way.

Speaker 2

29:33

It's called teaching to big ideas. So, at the moment, people have standards that have been written, and then textbooks have taken these standards and made not very good questions. And if you look at the standards, like I have some written down here, just reading the standards, it makes math seem really boring and uninspiring.

Speaker 3

29:57

LUKE HENRY What are the kind of, can you give a few examples? MARTA KRUPINSKA

Speaker 2

30:01

So this is an interesting example. In third grade, there are 3 different standards about unit squares.

Speaker 3

30:10

Okay.

Speaker 2

30:11

So this is 1 of them. A square with side length 1 unit called a unit square is said to have 1 square unit of area and can be used to measure area.

Speaker 3

30:20

And that's something you're expected to learn.

Speaker 2

30:23

That is something, so that's a standard. The textbook authors say, oh, I'm gonna make a question about that, and they translate the standards into narrow questions.

Speaker 3

30:31

And then you measure success by your ability to deliver on these standards.

Speaker 2

30:36

So the standards themselves, I think of maths, and many people think maths in this way, as a subject of like a few big ideas and really important connections between them. So you could think of it as like a network map of ideas and connections. And what standards do is they take that beautiful map and they chop it up like this into lots of little pieces and they deliver the pieces to schools.

Speaker 2

31:01

And so teachers don't see the connections between ideas, nor do the kids. So anyway, this is a bit of a long way of saying that what we've done in this new initiative is we have set out maths as a set of big ideas and connections between them. So this is grade 3. So instead of there being 60 standards, we've said, well, you can pull these different standards to get in with each other and also value the ways these are connected.

Speaker 3

31:37

And by the way, for people who are just listening, we're looking at a small number of like big concepts within mathematics, square tiles, measuring fraction, shape and time, and then how they're interconnected. And so the goal is for, this is for grade 3, for example.

Speaker 2

31:55

Yeah. And so we've set out for the state of California, the whole of Mathematics K10 as a set of big ideas and connections. So we know that teachers, it works really well if they say, okay, so a big idea in my grade is measuring. And instead of reading 5 procedural statements that involve measuring, they think, okay, measuring is a big idea.

Speaker 2

32:22

What rich, deep activity can I use that teaches measuring to kids? And as kids work on these deep, rich activities, maybe over a few days, turns out a lot of maths comes into it. So we're recommending that let's not teach maths according to all these multiple, multiple statements and lots and lots of short questions. Instead, let's teach maths by thinking about what are the big ideas and what are really rich, deep activities that teach those big ideas.

Speaker 3

32:53

So that's the, like how you teach it and maximize learning. What about like from a school district perspective, like measuring how well you're doing, grades and tests and stuff

Speaker 2

33:06

like that. Do you

Speaker 3

33:07

throw those out or is it possible? I'm not

Speaker 2

33:09

a fan of grades and tests myself. I think grades are fine if they're used at the end of a course. So at the end of my maths course, I might get a grade because a grade is meant to be a summative measure.

Speaker 2

33:24

It kind of describes your summative achievement. But the problem we have in maths classrooms across the US is people use grades all the time, every week or every day even. My own kids, when they went through high school, technology has not helped with this. When they went through high school, they knew they were being graded for everything they did, everything.

Speaker 2

33:46

And not only were they being graded for everything, but they could see it in the grade book online, and it would alter every class they went into. So this is the ultimate, what I think of as a performance culture. You're there to perform, somebody's measuring you, you see your score. So I think that's not conducive for deep learning.

Speaker 2

34:08

And yes, have a great at the end of the year. But during the year, you can assess kids in much better ways. Like teachers can, a great way of assessing kids is to give them a rubric that kind of outlines what they're learning over the course of a unit or a few weeks. So kids can actually see the journey they're on, like this is what we're doing mathematically.

Speaker 2

34:31

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

Speaker 2

35:02

At the end of the year, sure, they can have a grade. But during the year, they get these much more informative measures.

Speaker 3

35:10

I do think this might be more for college, but Maybe not. Some of the best classes I've had is when I got a special, like set aside, like the professor clearly saw that I was interested in some aspect of a thing. And then I've a few in mind and 1 in particular, he said that he kind of challenged me.

Speaker 3

35:38

So this is outside of grades and all that kind of stuff that basically it's like reverse psychology. I don't think this can be done. And so I gave everything to do that particular thing. So this was, happened to be in an artificial intelligence class.

Speaker 3

35:54

But I think that like special treatment of taking students who are especially like excellent at a particular little aspect that you see their eyes light up. I often think like, maybe it's tempting for a teacher to think you've already succeeded there, but they're actually signaling to you that like you could really launch them on their way. Yeah. And I don't know, that's too much to expect from teachers, I think, to pay attention to all of that, because it's really difficult.

Speaker 3

36:25

But I just kind of remember who are the biggest, the most important people in the history of my life, of education, and it's those people that who really didn't just like inspire me with their awesomeness, which they did, but also just, they pushed me a little. Like, they gave me a little push. And that requires focusing on the quote unquote excellent students in the class.

Speaker 2

36:50

Yeah, I think what's important though is teachers to have the perspective that they don't know who's gonna be excellent at something before they give out the activity. Exactly. And in our camp classes that we ran, sometimes students would finish ahead of other students.

Speaker 2

37:10

And we would say to them, can you write a question that's like this but different? And over time, we encouraged them to like extend things further. I remember we were doing 1 activity where kids were working out the borders of a square and how big this border would be in different case sizes. And 1 of the boys came up at the end of the class and said, I've been thinking about how you do this with the Pentagon.

Speaker 2

37:40

And I said, that's fantastic. What does it look like with the Pentagon? Go find out, see if you can discover. So I didn't know he was gonna come up and say that, and I didn't have in my head, like this is the kid who could have this extension task, but you can still do that as a teacher.

Speaker 2

37:59

When kids get excited about something or they're doing well in something, have them extend it, go further. It's great.

Speaker 3

38:06

And then you also, like, this is like teacher and coach, you could say it in different ways to different students. Like for me, the right thing to say is almost to say, I don't think you could do this, this is too hard. Like that's what I need to hear.

Speaker 3

38:22

Because it's like, no, there's an immediate push. But with some people, if they're a little bit more, I mean, it all has to do with upbringing, just how your genetics is. They might be much more, that might break them.

Speaker 2

38:35

Yeah, that might break them. And so

Speaker 3

38:36

you have to be also sensitive to that. I mean, teaching is really difficult.

Speaker 2

38:39

It is really difficult.

Speaker 3

38:41

For this very reason.

Speaker 2

38:42

It is.

Speaker 3

38:43

So what is the best way to teach math, to learn math, at those early few days when you just want to capture them?

Speaker 2

38:53

I do something, actually there's a video of me doing this on our website, that I love when I first meet students. And this is what I do. I show them a picture.

Speaker 2

39:05

This is the picture I show them. And it's a picture of 7 dots like this. And I show it for just a few seconds. And I say to them, I'd like you to tell me how many dots there are, but I don't need to count them.

Speaker 2

39:19

I want you to group the dots. And I show it to them and then I take it away before they've even had enough time to count them. And then I ask them, so how did you see it? And I go around the room and amazingly enough, there's probably 18 different ways of seeing these 7 dots.

Speaker 2

39:39

And so I ask people, tell me how you grouped it. And some people see it as like an outside hole with a center dot. Some people see like stripes of lines. Some people see segments.

Speaker 2

39:51

And I collect them all and I put them on the board and at the end I say, look at this. We are a class of 30 kids and we saw these 7 dots in 18 different ways. There's actually a mathematical term for this. It's called groupitizing.

Speaker 3

40:03

Groupitizing? Yeah. I like it.

Speaker 2

40:06

It's kind of cool. So turns out though, that how well you groupitize predicts how well you do in math.

Speaker 3

40:15

Is it a raw talent or is it just something that you can develop? I don't

Speaker 2

40:19

think it's, I don't think you're born groupitizing, I think, but some kids have developed that ability, if you like, and you can learn it. So this, to me, is part of how wrong we have maths, that we think to tell whether a kid's good at maths, we're going to give them a speed test on multiples. But actually, seeing how kids group dots could be a more important assessment of how well they're gonna do in math.

Speaker 2

40:48

Anyway, I diverge. What I like to do when I start off with kids is show them, I'm gonna give you math problems, I'm gonna value the different ways you see them. And it turns out you can do this kind of problem asking people how they group dots with young children or with graduate students. And it's engaging for all of them.

Speaker 3

41:10

You talk about creativity a little bit and flexibility in your book Limitless. What's the role of that? So it sounds like there's a bit of that kind of thing involved in groupitizing.

Speaker 3

41:22

Yeah, yeah. I love this term. So what would you say is the role of creativity and flexibility in the learning of math?

Speaker 2

41:31

I think what we know now is that what we need for this 21st century world we live in is a flexible mind. School should not really be about teaching kids particular methods, but teaching them to approach problems with flexibility. Being creative, thinking creatively is really important.