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This is the Lex Friedman Podcast, and here is my conversation with Joe Bowler.
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What to you is beautiful about mathematics?
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I love a mathematics that some people don t even think of as mathematics, which is beautiful, creative.
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mathematics, where we look at maths in different ways, we visualize it, we think about different solutions to problems.
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A lot of people think of maths as you have one method and one answer.
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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.
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So that is what is beautiful to me about mathematics, that you can see and solve it in many different ways.
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Also the sad part that many people think that maths is just one answer and one method.
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So to you, the beauty emerges when you have a problem with a solution and you start adding other solutions.
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Simpler solutions, weirder solutions, more interesting, some that are visual, some that are algebraic, geometry, all that kind of stuff.
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Yeah, I mean, I always say that you can take any maths area and make it visual.
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And we say to teachers, give us your most dry, boring maths, and we ll make it a visual, interesting, creative problem.
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And it turns out you can do that with any area of maths.
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And I think it s been a great disservice to kids and others that it s always been numbers, lots and lots of numbers.
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Numbers can be great, but you can think about maths in other ways besides numbers.
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Do you find that most people are better visual learners, or is this just something that s complementary?
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What s the kind of the full spectrum of students and the way they like to explore math, would you say?
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There s definitely people who come into the classes I do who are more interested in visual thinking and like visual approaches, but it turns out what the neuroscience is telling us is that when we think about maths, there are two visual pathways in the brain,
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we should all be thinking about it visually.
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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.
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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.
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So you were saying that there s some kind of interconnected aspect to it, so the visual connects with the non-visual.
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Yeah, so this is what the neuroscience has shown us that when you work on a maths problem, there are five different brain pathways, and that the most high achieving people in the world are people who have more connections between these pathways.
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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.
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And if you maybe write about it with words, that would cause another connection.
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Or maybe you build it with something physical, that would cause a different connection.
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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.
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That will cause that great connected brain.
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You know, there s these stories of physicists doing the same.
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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.
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And for that, you have to intuit your way there.
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And you mentioned offline that one of the ways you might approach a problem is to try to tell a story about it.
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And some of it is like legend, but I m sure it s not always.
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You have Einstein thinking about a train and the speed of light, and that kind of intuition is useful.
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You start to imagine a physical world.
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How does this idea manifest itself in the physical world?
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And then start playing in your mind with that physical world and think, is this going to be true?
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Is this going to be true?
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Right.
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Einstein is well known for thinking visually.
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And people talk about how he really didn t want to go anywhere with problems without thinking about them visually.
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But the other thing you mentioned that sparked something for me is thinking with intuition, like having intuition about math problems.
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That s another thing that s often absent in math class, the idea that you might think about a problem and use your intuition.
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But so important.
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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.
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Yeah, I mean, that s what it is.
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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?
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In order to understand the pattern, you might want to then start utilizing visualization.
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But ultimately, that s all in service of like,
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solving the puzzle, 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.
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But it s hard, it s hard.
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Reasoning is really hard.
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Yeah, it s hard.
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I mean, I love to value what s hard in maths instead of being afraid of it.
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We know that when you struggle, that s actually a really good time for your brain.
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You want to be struggling when you re thinking about things.
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So if it s hard to think intuitively about something, that s probably a really good
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time for your brain.
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I used to work with somebody called Sebastian Thrun, who is a great sort of mathematician, you might think of him, AI person.
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And I remember in one interview I did with him, he talked about how they d built robots, I think for the Smithsonian, and how they were having this trouble with them picking up white noise.
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And he said they had to solve it, they had to work out what s going on, and how he intuitively
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worked out what the problem was.
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But then it took him three weeks to show it mathematically.
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I thought that was really interesting that how you can have this intuition and know something works.
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It s kind of different from going through that long mathematical process of proving it, but so important.
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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.
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You see that with Feynman and his, I mean, it just, all of these physicists, definitely you see,
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starting with intuition, sometimes starting with an experiment, and then the experiment inspires intuition.
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But you can think of an experiment as a kind of visualization.
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Just like let s take whatever the heck we re looking at and draw it, and draw the pattern as it evolves, as the thing grows.
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for n equals one, for n equals two, n equals three, and you start to play with it.
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And then in the modern day, which I loved doing, is you can write a program that then visualizes it for you.
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And then you can start exploring it programmatically.
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And then...
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you can do so interactively too.
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I tend to not like interactive because it takes way too much work.
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You have to click and move and stuff.
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I love to interact through writing programs.
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That s my particular brain, software engineer.
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So you can do all these kinds of visualizations.
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And then there s the tools of visualization like color,
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all those kinds of things.
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You re absolutely right.
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They re actually not taught very much.
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Like the art of visualization.
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Not taught.
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And we love as well color coding.
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Like when you represent something mathematically, you can show color to show the growth and kind of code that.
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So if I have an algebraic expression for a pattern, maybe I show the X with a certain color, but also write in that color so you can
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see the relationship.
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Very cool.
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And, um, yeah, we particularly in our work with elementary teachers, many of them come to our workshops, and they re literally in tears when they see things making sense visually, because they ve spent their whole lives and not realizing you can really understand things with these visuals.
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It s quite powerful.
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You say that there s something valuable to learning when the thing that you re doing is challenging, is difficult.
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So a lot of people say math is hard or math is too hard or too hard for me.
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Do you think math should be easy or should it be hard?
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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.
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And I think the problem in education is a lot of people have got this idea that you re either born with a maths brain or you re not.
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So when they start to struggle, they think, Oh, I don t have that maths brain.
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And then they will literally sort of switch off in their brain and things will go downhill from that point.
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So struggle becomes a lot easier.
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And you re able to struggle if you don t have that idea.
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But you know that you you can do it, you have to go through this struggle to get there.
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But you re, you re able to do that.
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And so we re hampered in being able to struggle with these ideas we ve been given about what we can do.
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Can I ask a difficult question here?
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So there s kind of, I don t know what the right term is, but some people struggle with learning in different ways.
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Like their brain is constructed in different ways.
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And how much should, as educators, should we make room for that?
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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.
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I can t learn it this way.
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How do you find that line?
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How do you operate in that gray area?
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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.
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But this is also why I believe it s so important to have this multidimensional approach to maths.
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We ve really offered it in one way, which is
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Here s some numbers and a method, you follow me, do what I just did, and then reproduce it.
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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.
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But when you open up maths and you let kids experience it in different ways, maybe visually, with numbers, with words,
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What happens is kids, there are many more kids who can access it.
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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 one of the reasons we want to open it up so that there are different ways of accessing it.
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And then that s not really a problem.
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So I grew up in the Soviet Union and fell in love with math early.
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I was forced into math early and fell in love through force.
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That s good.
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Well, good that you fell in love.
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But something we talked about a little bit is there s such
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a value for excellence.
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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.
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And it s not improving, it s like being really good.
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I mean, we are much more like that with sports, for example.
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It s understood, you re going to star on the basketball team if you re going to be better than the other guys, the other girls on the team.
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Coupled with the belief this could be partially a communist belief.
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I don t know but the belief that everybody is capable of being great But if you re not great, that s your fault and you need to work harder and I remember I had a sense that Probably delusional but I could win a Nobel Prize.
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I don t even know what that entails But I thought
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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.
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And I remember early in my life thinking, I m going to invent the time machine.
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And like the tools of mathematics were in service of that dream of winning the Nobel Prize.
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It s silly.
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I didn t really think in those concrete terms, but I just thought I could be great, that feeling.
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And then when you struggle, the belief that you could be great is like struggle is good.
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Right, pushes you on, yeah.
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And so the other thing about the Soviet system that I d love to hear your comments about is just the sheer hours of math.
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The number of courses, you re talking about a lot of geometry, a lot more geometry.
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I think in the American system, you take maybe one year of geometry.
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In high school, yeah.
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In high school.
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First of all, geometry is beautiful, it s visual.
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And then you get to reason through proofs and stuff like that.
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In Russia, I remember just being nailed over and over, it was just nonstop.
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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.
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So math, but also science and literature, by the way,
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was also hit very hard.
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We read a lot of serious adult stuff.
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America does that a little bit too.
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They challenge young adults with good literature, but they don t challenge adults very much with math.
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So those two things, valuing excellence and just a lot of math in the curriculum.
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Do you find that interesting?
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Because it seems to have been successful.
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Yeah, I think that s very interesting.
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There is a lot of success of people coming through the Soviet system.
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I think something that s very different to the U.S.
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and other countries in the world is that idea that excellence is important and you can get there if you work hard.
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In the U.S., there s an idea that excellence is important, but then kids are given the idea in many ways that you can either do it or you re one of the people who can t.
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So many students in the school system think they re one of the kids who can t.
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So there s no point in trying hard because you re never going to get there.
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So if you can switch that idea, it would be huge.
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And it seems from what you ve said that in the Soviet Union, that idea is really different.
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Now, the downside of that idea that anybody can get there if you work hard,
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is that thought that if you re not getting there, it s your fault.
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And I would add something into that.
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I would say that anybody can get there, but they need to work hard and they also need good teaching.
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Because there are some people who really can t get there because they re not given access to that good teaching.
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So, but that would be huge, that change.
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As to doing lots of maths, if,
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If maths was interesting and open and creative and multidimensional, I would be all for it.
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We actually run summer camps at Stanford where we invite kids in and we give them this maths that I love.
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And in our camp classrooms, they were three hours long.
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And
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When we were planning, the teachers were like, three hours, are we going to be able to keep the kids excited for three hours?
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Turned out, they didn t want to go to break or lunch.
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They d be so into these mathematical patterns.
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We couldn t stop them.
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It was amazing.
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So yeah, if maths was more like that, then I think having more of it would be a really good thing.
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So what age are you talking about?
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Could you comment on what age is the most important when people quit math or give up on themselves or on math in general?
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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.
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I think a lot of kids start to give up on themselves and maths from about fifth grade.
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And then those middle school years are really important.
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And fifth grade can be pivotal for kids just because they re allowed to explore and
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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.
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And that s when kids start to feel really badly about themselves.
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And so middle school years, our camps are middle school students.
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We think of those years as really pivotal.
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Many kids in those years are deciding, yes, I m going to keep going with STEM subjects, or no, I m not, that this isn t for me.
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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.
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So what s the role of the teacher in this?
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So one is the explanation of the subject, but do you think teachers should almost do like one-on-one, you know, little Johnny, I believe in you kind of thing?
1
SPEAKER_00
Like that energy of like... Turns out it s really important.
1
SPEAKER_00
There s a study that was done, it was actually done in high school English classrooms.
1
SPEAKER_00
where all kids wrote an essay for their teacher.
1
SPEAKER_00
And this was done as an experiment.
1
SPEAKER_00
Half of the kids got feedback from their teacher, diagnostic feedback, which is great.
1
SPEAKER_00
But for half of the kids, it said an extra sentence at the bottom that the researchers had put on.
1
SPEAKER_00
And the kids who read that extra sentence did significantly better in English a whole year later.
1
SPEAKER_00
The only change was this one sentence.
1
SPEAKER_01
What did the sentence say?
1
SPEAKER_00
So, what did the sentence say?
1
SPEAKER_00
The sentence said, I m giving you this feedback because I believe in you.
1
SPEAKER_00
And the kids who read that did better a year later.
1
SPEAKER_00
So when I share this with teachers, I say, you know, I m not suggesting you put on the bottom of all kids  work.
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SPEAKER_00
I m giving this feedback because I believe in you.
1
SPEAKER_00
One of the teachers said to me, we don t put it on a stamp.
1
SPEAKER_00
I said, no, don t put it on a stamp.
1
SPEAKER_00
But your words are really important.
1
SPEAKER_00
And
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SPEAKER_00
Kids are sitting in classrooms all the time thinking, what does my teacher think of me?
1
SPEAKER_00
Does my teacher think I can do this?
1
SPEAKER_00
So it turns out it is really important to be saying to kids, I know you can do this.
1
SPEAKER_00
And those messages are not given enough by teachers.
1
SPEAKER_01
And really believe it.
1
SPEAKER_00
And believe it.
1
SPEAKER_00
Yeah, it s like, you can t just say it, you have to believe it.
1
SPEAKER_01
I sometimes because like,
1
SPEAKER_01
It s such a funny dance, because I m such a perfectionist, and I m extremely self-critical, and I have one of the students come up to me, and it s clear to me that they re not even close to good.
1
SPEAKER_01
And it s tempting for me to be like, to sort of give up on them mentally.
1
SPEAKER_01
But the reality is, if you look at many great people throughout history,
1
SPEAKER_01
they sucked at some point.
1
SPEAKER_01
And some of the greatest took non-linear paths to where they sucked for long into later life.
1
SPEAKER_01
And so always kind of believing that this person can be great.
1
SPEAKER_01
You have to communicate that plus the fact that they have to work hard.
1
SPEAKER_00
Yeah.
1
SPEAKER_00
Yeah.
1
SPEAKER_00
And you re right.
1
SPEAKER_00
Silicon Valley, where I live, is filled with people who are dropouts of school or who had special needs, who didn t succeed.
1
SPEAKER_00
It s very interesting that have gone on to do amazing work in creative ways.
1
SPEAKER_00
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.
1
SPEAKER_00
often slower thinkers, they think slowly and deeply.
1
SPEAKER_00
And they often get the idea early on that they can t be good at maths or other subjects.
1
SPEAKER_00
So, yeah, I think many of those subject people are the ones who go on and do amazing things.
1
SPEAKER_01
So there s a guy named Eric Weinstein.
1
SPEAKER_01
I know many mathematicians like this, but he talks a lot about having a non-standard way of learning.
1
SPEAKER_01
I mean, a lot of great mathematicians, a lot of great physicists are like that.
1
SPEAKER_01
And he felt like he became quickly, he got his PhD at Harvard, became quickly an outcast of the system.
1
SPEAKER_01
The education, especially early education system didn t help him.
1
SPEAKER_01
Is there ways for an education system to support people like that?
1
SPEAKER_01
Is it this kind of multi-dimensional learning that you re mentioning?
1
SPEAKER_00
Absolutely, absolutely.
1
SPEAKER_00
I mean, I think our education system still uses an approach that was in classrooms hundreds of years ago.
1
SPEAKER_00
The textbooks have a lot to answer for in producing this very uninspiring mathematics
1
SPEAKER_00
But yeah, if you open up the subject and have people see and solve it in different ways and value those different ways.
1
SPEAKER_00
Somebody I appreciated a lot is a mathematician called Mary Mizzikani.
1
SPEAKER_00
I don t know if you ve heard of her.
1
SPEAKER_00
She won the Fields Medal.
1
SPEAKER_00
She was from Iran.
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SPEAKER_00
First woman in the world to win the Fields Medal in mathematics.
1
SPEAKER_00
She died when she was 40.
1
SPEAKER_00
She was at Stanford.
1
SPEAKER_00
But her work was entirely visual.
1
SPEAKER_00
And she talked about how her daughter thought she was an artist because she was always visualizing.
1
SPEAKER_00
I attended, she asked me to chair the PhD defense for one of her students.
1
SPEAKER_00
And I went to the defense in the math department.
1
SPEAKER_00
And it was so interesting because this young woman spent like two hours sharing her work, all of it was visual.
1
SPEAKER_00
In fact, I don t think I saw any numbers at all.
1
SPEAKER_00
And I remember that day thinking, wow, I could have brought her like 13 year old into this PhD defense, they would not recognize this as maths.
1
SPEAKER_00
But when Maryam Izzakhani won the Fields Medal, all these other mathematicians were saying that her work had connected all these previously unconnected areas of maths.
1
SPEAKER_00
She also shared that when she was in school, when she was about 13, she was told that she couldn t do maths.
1
SPEAKER_00
She was told that by her teacher.
1
SPEAKER_01
This is Iran?
1
SPEAKER_01
In Iran, yeah.
1
SPEAKER_00
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.
1
SPEAKER_01
I ve been told by a lot of people in my life that I can t do something.
1
SPEAKER_01
I m very definitely non-standard.
1
SPEAKER_01
But all it takes, that s why people talk about the one teacher that changed everything for them.
1
SPEAKER_01
All it takes is one teacher.
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SPEAKER_01
That s the power of that.
1
SPEAKER_01
So that should be inspiring to teachers.
1
SPEAKER_00
I think it is.
1
SPEAKER_01
You as a single person, given the education system, given the incentives, you have the power to truly change lives.
1
SPEAKER_01
And like 20 years from now, I feel as a medalist will walk up to you and say thank you.
1
SPEAKER_00
Yeah, you did that for me.
1
SPEAKER_00
Yeah, absolutely.
1
SPEAKER_00
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  maths relationship or other subjects and forever.
1
SPEAKER_01
What s the role of the parents in this picture?
1
SPEAKER_01
Let s go to another difficult subject.
1
SPEAKER_00
Yeah, that is a difficult subject.
1
SPEAKER_00
One study found that the amount of maths anxiety parents had predicted their child s achievement in school, but only if they helped with homework.
1
SPEAKER_00
So...
1
SPEAKER_01
Oh, that s so funny.
1
SPEAKER_00
Yeah, there are some interesting implications for this.
1
SPEAKER_00
I mean, you can see how it works.
1
SPEAKER_00
If you have maths anxiety and you re helping your kids with homework, you re probably communicating things like, I was terrible at this at school, and that s how it gets passed on to kids.
1
SPEAKER_00
So one implication is if you have a really bad relationship with maths, you hate maths, you have maths anxiety, just don t do maths homework with your kids.
1
SPEAKER_00
But we have, on our website, we have a little sheet for parents of ways to interact around maths with your kids.
1
SPEAKER_01
And... That s ucubed.org?
1
SPEAKER_01
That s ucubed.org, yes.
1
SPEAKER_00
So one of the things I say to parents when I give parent presentations is, even if you hate maths, you need to just fake it with your kids.
1
SPEAKER_00
You should be always endlessly optimistic and happy about doing maths.
1
SPEAKER_01
And... I m always curious about this, so I hope to have kids one day.
1
SPEAKER_01
I don t have kids currently.
1
SPEAKER_01
Are parents okay with sucking at math and then trying to get their kid to be better than them, essentially?
1
SPEAKER_01
Is that a difficult thing for a lot of parents?
1
SPEAKER_00
It is difficult.
1
SPEAKER_01
To have, it s almost like an ego thing.
1
SPEAKER_01
I never got good at this and I probably should have.
1
SPEAKER_01
And...
1
SPEAKER_01
Yeah, I mean, to me, 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.
1
SPEAKER_01
So do you experience the same with parents?
1
SPEAKER_00
I think, I mean, I haven t experienced parents worrying that their kids will be better than them.
1
SPEAKER_00
I have experienced parents
1
SPEAKER_00
just having a really bad relationship with maths and not wanting to help, not knowing how to help, saying things.
1
SPEAKER_00
Another study showed that when mothers say to their daughters, I was bad at maths in school, their daughter s achievement goes down.
1
SPEAKER_00
So we know that kids pick up on these messages, which is why I say you should fake it.
1
SPEAKER_00
But also, I know that lots of people have just had a really bad relationship with maths, even successful people.
1
SPEAKER_00
The undergrads I teach at Stanford have pretty much always done well in maths.
1
SPEAKER_00
But they come to Stanford thinking maths is a set of methods to memorize.
1
SPEAKER_00
And so do many parents believe that.
1
SPEAKER_00
There s one method that you memorize and then you reproduce it.
1
SPEAKER_00
So until people have really had an experience of what I think of as the other maths, 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.
1
SPEAKER_01
Is there for a teacher, if we re to like systematize it, is there something teachers can do to do this more effectively?
1
SPEAKER_01
So you mentioned the textbook.
1
SPEAKER_01
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?
1
SPEAKER_00
So I do think there s a way of teaching maths that changes everything for kids and teachers.
1
SPEAKER_00
So I m one of five writers of a new framework for the state of California, a new maths framework.
1
SPEAKER_00
It s coming out next year.
1
SPEAKER_00
And we are recommending through this maths framework that people teach in this way.
1
SPEAKER_00
It s called teaching to big ideas.
1
SPEAKER_00
So at the moment, people have standards that have been written, and then textbooks have taken these standards and made not very good questions.
1
SPEAKER_00
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.
1
SPEAKER_01
What are the kind of, can you give a few examples?
1
SPEAKER_00
So this is an interesting example.
1
SPEAKER_00
In third grade, there are three different standards about unit squares.
1
SPEAKER_00
Okay.
1
SPEAKER_00
So this is one of them.
1
SPEAKER_00
A square with side length one unit, called a unit square, is said to have one square unit of area and can be used to measure area.
1
SPEAKER_01
And that s something you re expected to learn.
1
SPEAKER_00
That is something, so that s a standard.
1
SPEAKER_00
The textbook authors say, oh, I m gonna make a question about that, and they translate the standards into narrow questions.
1
SPEAKER_01
And then you measure success by your ability to deliver on these standards.
1
SPEAKER_00
So the standards themselves, I think of maths, and many people think of maths in this way, as a subject of a few big ideas and really important connections between them.
1
SPEAKER_00
So you can think of it as like a network map of ideas and connections.
1
SPEAKER_00
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.
1
SPEAKER_00
So teachers don t see the connections between ideas, nor do the kids.
1
SPEAKER_00
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.
1
SPEAKER_00
So this is a grade three.
1
SPEAKER_00
So instead of there being six
2
SPEAKER_00
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.
2
SPEAKER_01
And by the way, for people who are just listening, we re looking at a small number of big concepts within mathematics, square tiles, measuring, fraction, shape, and time, and then how they re interconnected.
2
SPEAKER_01
And so the goal, and this is for grade three, for example.
2
SPEAKER_00
Yeah, so we ve set out for the state of California the whole of mathematics K10 as a set of big ideas and connections.
2
SPEAKER_00
So we know that teachers, it works really well if they say, okay, so a big idea in my grade is measuring.
2
SPEAKER_00
And instead of reading five procedural statements that involve measuring, they think, okay, measuring is a big idea, what rich, deep activity can I use that teaches measuring to kids?
2
SPEAKER_00
And as kids work on these deep, rich activities, maybe over a few days, turns out a lot of maths comes into it.
2
SPEAKER_00
So we re recommending that let s not teach maths according to all these multiple, multiple statements, and lots and lots of short questions.
2
SPEAKER_00
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.
2
SPEAKER_01
So that s the, like how you teach it and maximize learning.
2
SPEAKER_01
What about like from a school district perspective, like measuring how well you re doing, you know, grades and tests and stuff like that.
2
SPEAKER_01
Do you throw those out or is it possible?
2
SPEAKER_00
I m not a fan of grades and tests myself.
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SPEAKER_00
I think,
2
SPEAKER_00
grades are fine if they re used at the end of a course.
2
SPEAKER_00
So at the end of my maths course, I might get a grade because a grade is meant to be a summative measure, it kind of describes your summative achievement.
2
SPEAKER_00
But the problem we have in maths classrooms across the US is people use grades all the time, every week, or every day, even.
2
SPEAKER_00
My own kids, when they went through high school, technology has not helped with this.
2
SPEAKER_00
When they went through high school, they knew they were being graded for everything they did, everything.
2
SPEAKER_00
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.
2
SPEAKER_00
So this is the ultimate, what I think of as a performance culture.
2
SPEAKER_00
You re there to perform, somebody s measuring you, you see your score.
2
SPEAKER_00
So I think that s not conducive for deep learning.
2
SPEAKER_00
And yes, have a great at the end of the year.
2
SPEAKER_00
But during the year, you can assess kids in much better ways.
2
SPEAKER_00
Like teachers can, a great way of assessing kids is to give them a rubric.
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SPEAKER_00
that kind of outlines what they re learning over the course of a unit or a few weeks.
2
SPEAKER_00
So kids can actually see the journey they re on, like this is what we re doing mathematically.
2
SPEAKER_00
Sometimes they self-assess on those units.
2
SPEAKER_00
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.
2
SPEAKER_00
So instead of kids just thinking, 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.
2
SPEAKER_00
And they re involved in the process of knowing where they are mathematically.
2
SPEAKER_00
At the end of the year, sure, they can have a grade.
2
SPEAKER_00
But during the year, they get these much more informative measures.
2
SPEAKER_01
I do think this might be more for college, but maybe not.
2
SPEAKER_01
Some of the best classes I ve had is when I got a special set aside.
2
SPEAKER_01
The professor clearly saw that I was interested in some aspect of a thing.
2
SPEAKER_01
And then I have a few in mind and one in particular, and he said that he kind of challenged me, so this is outside of grades and all that kind of stuff, that
2
SPEAKER_01
Basically, it s like reverse psychology.
2
SPEAKER_01
I don t think this can be done.
2
SPEAKER_01
And so I gave everything to do that particular thing.
2
SPEAKER_01
So this happened to be in an artificial intelligence class.
2
SPEAKER_01
But I think that special treatment of taking students who are especially excellent at a particular little aspect, that you see their eyes light up,
2
SPEAKER_01
I often think maybe it s tempting for a teacher to think you ve already succeeded there, but they re actually signaling to you that you could really launch them on their way.
2
SPEAKER_01
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.
2
SPEAKER_01
But I just kind of remember who are the biggest,
2
SPEAKER_01
the most important people in the history of my life of education.
2
SPEAKER_01
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.
2
SPEAKER_01
And that requires focusing on the quote unquote excellent students in the class.
2
SPEAKER_00
I think what s important, though, is teachers to have the perspective that they don t know who s going to be excellent at something before they give out the activity.
2
SPEAKER_00
And in our camp classes that we ran, sometimes students would finish ahead of other students.
2
SPEAKER_00
And we would say to them, can you write a question that s like this but different?
2
SPEAKER_00
And over time, we encouraged them to like extend things further.
2
SPEAKER_00
I remember we were doing one activity where kids were working out the borders of a square and how big this border would be in different case sizes.
2
SPEAKER_00
And one of the boys came up at the end of the class and said, I ve been thinking about how you do this with the Pentagon.
2
SPEAKER_00
And I said, that s fantastic.
2
SPEAKER_00
What does it look like with Pentagon?
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SPEAKER_00
Go find out, see if you can discover.
2
SPEAKER_00
So I didn t know he was gonna come up and say that.
2
SPEAKER_00
And I didn t have in my head, this is the kid who could have this extension task.
2
SPEAKER_00
But you can still do that as a teacher.
2
SPEAKER_00
When kids get excited about something or they re doing well in something, have them extend it, go further.
2
SPEAKER_00
It s great.
2
SPEAKER_01
And then you also, like, this is like teacher and coach, you could say it in different ways to different students.
2
SPEAKER_01
Like for me, the right thing to say is almost to say, I don t think you could do this.
2
SPEAKER_01
This is too hard.
2
SPEAKER_01
Like, that s what I need to hear.
2
SPEAKER_01
Cause it s like, no, I, you know, there s an immediate push, but with some people, if they re a little bit more, I mean, it s all has to do with upbringing, how your genetics is, they might be much more, that might break them.
2
SPEAKER_00
Yeah, that might break them.
2
SPEAKER_01
So you have to be also sensitive to that.
2
SPEAKER_01
I mean, teaching is really difficult.
2
SPEAKER_01
It is really difficult.
2
SPEAKER_01
For this very reason.
2
SPEAKER_01
It is.
2
SPEAKER_01
So what is the best way to teach math, to learn math at those early few days when you just want to capture them?
2
SPEAKER_00
I do something.
2
SPEAKER_00
Actually, there s a video of me doing this on our website that I love when I first meet students.
2
SPEAKER_00
And this is what I do.
2
SPEAKER_00
I show them a picture.
2
SPEAKER_00
This is the picture I show them.
2
SPEAKER_00
And it s a picture of seven dots like this.
2
SPEAKER_00
And I show it for just a few seconds.
2
SPEAKER_00
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.
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SPEAKER_00
I m going to group the dots.
2
SPEAKER_00
And I show it them and then I, I take it away before they ve even had enough time to count them.
2
SPEAKER_00
And then I asked them, so how did you see it?
2
SPEAKER_00
And I go around the room and
2
SPEAKER_00
Amazingly enough, there s probably 18 different ways of seeing these seven dots.
2
SPEAKER_00
And so I asked people, tell me how you grouped it.
2
SPEAKER_00
And some people see it as like an outside hole with a center dot.
2
SPEAKER_00
Some people see like stripes of lines.
2
SPEAKER_00
Some people see segments.
2
SPEAKER_00
And I collect them all and I put them on the board and at the end I say, look at this, we are a class of 30 kids and we saw these seven dots in 18 different ways.
2
SPEAKER_00
There s actually a mathematical term for this, it s called groupitizing.
2
SPEAKER_01
Groupitizing?
2
SPEAKER_00
Yeah.
2
SPEAKER_01
I like it.
2
SPEAKER_00
It s kind of cool.
2
SPEAKER_00
So turns out though, that how well you groupitize predicts how well you do in maths.
2
SPEAKER_01
Is it a raw talent or is it just something that you can develop?
2
SPEAKER_00
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.
2
SPEAKER_00
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 gonna give them a speed test on multiples.
2
SPEAKER_00
But actually,
2
SPEAKER_00
seeing how kids group dots could be a more important assessment of how well they re going to do in math.
2
SPEAKER_00
Anyway, I diverge.
2
SPEAKER_00
What I like to do when I start off with kids is show them, I m going to give you math problems, I m going to value the different ways you see them.
2
SPEAKER_00
And turns out you can do this kind of problem asking people how they group dots with young children or with graduate students.
2
SPEAKER_00
And it s engaging for all of them.
2
SPEAKER_01
is you talk about creativity a little bit and flexibility in your book, Limitless.
2
SPEAKER_01
What s the role of that?
2
SPEAKER_01
So it sounds like there s a bit of that kind of thing involved in groupitizing.
2
SPEAKER_01
I love this term.
2
SPEAKER_01
So what would you say is the role of creativity and flexibility in the learning of math?
2
SPEAKER_00
I think what we know now is that what we need for this 21st century world we live in
2
SPEAKER_00
is a flexible mind.
2
SPEAKER_00
School should not really be about teaching kids particular methods, but teaching them to approach problems with flexibility.
2
SPEAKER_00
Being creative, thinking creatively is really important.
2
SPEAKER_00
So people don t think the words maths and creativity come together, but that s what I love about maths is the creative different ways you can see it.
2
SPEAKER_00
And so helping our kids
2
SPEAKER_00
There s a book I like a lot by been by physicists.
2
SPEAKER_00
You probably know this book called elastic.
2
SPEAKER_00
We might know it.
2
SPEAKER_00
And it s about how we want elastic minds, same kind of thing, flexible, creative minds.
2
SPEAKER_00
And schools do very little on developing that kind of mind.
2
SPEAKER_00
They do a lot of developing the kind of mind that a computer now does for us.
2
SPEAKER_01
Memorization.
2
SPEAKER_00
Memorization, doing procedures, a lot of things that we spend a lot of time in school on.
2
SPEAKER_00
In the world, when kids leave school, a computer will do that.
2
SPEAKER_00
And better than they will.
2
SPEAKER_00
But that creative, flexible thinking, we re kind of at ground zero at computers being able to engage in that thinking.
2
SPEAKER_00
Maybe we re a little above ground zero, but the human brain is perfectly suited for that creative, flexible thinking.
2
SPEAKER_00
That s what humans are so great at.
2
SPEAKER_00
So I would like the balance to shift in schools.
2
SPEAKER_00
Maybe you still need to do some procedural kind of thinking, but there should be a lot more of that creative, flexible thinking.
2
SPEAKER_01
And what s the role of other humans in this picture?
2
SPEAKER_01
So collaborative learning, so brainstorming together, so creativity as it emerges from the collective intelligence of multiple humans.
2
SPEAKER_00
Yeah, super important.
2
SPEAKER_00
And we know that also helps develop your brain, that social side of thinking.
2
SPEAKER_00
And I love mathematics collaboration, where people build on each other s ideas, and they come up with amazing things.
2
SPEAKER_00
I actually taught 100 students calculus at Stanford recently, undergrads, and we taught them to collaborate.
2
SPEAKER_00
So these students came in Stanford and most of them were against collaboration in math.
2
SPEAKER_01
This is before COVID in person?
2
SPEAKER_00
Yeah, it was just before COVID hit, it was 2019.
2
SPEAKER_00
And this summer- Sorry, you said they re against?
2
SPEAKER_01
Yeah, so-
2
SPEAKER_00
It s really interesting.
2
SPEAKER_00
So they d only experienced maths individually, in a kind of competitive individual way.
2
SPEAKER_00
And if they had experienced it as group work, it had been a bad experience.
2
SPEAKER_00
Like maybe they were the one who did it all and the others didn t do much.
2
SPEAKER_00
So they were kind of against collaboration.
2
SPEAKER_00
They didn t see any role for it in maths.
2
SPEAKER_00
And we taught them to collaborate.
2
SPEAKER_00
And it was hard work because as well as the fact that they were kind of against collaboration, they came in with a lot of like social comparison thinking.
2
SPEAKER_00
So I m in this room with other Stanford undergrads and they re better than me.
2
SPEAKER_00
So when we set them to work on a math problem together, the first one was kind of a disaster because they were all like, they re better than me, they re faster than me.
2
SPEAKER_00
They came up with something I didn t come up with.
2
SPEAKER_00
So we taught them.
2
SPEAKER_00
to let go of that thinking and to work well together.
2
SPEAKER_00
And one of the things we did, we decided we wanted to do a pre and post test at the end of this teaching, it was only four weeks long.
2
SPEAKER_00
But we knew we didn t want to give them like a time test of individual work.
2
SPEAKER_00
So we gave them
2
SPEAKER_00
an applied problem to do at the beginning.
2
SPEAKER_00
And we gave them to do in pairs together.
2
SPEAKER_00
And we gave each of them a different colored pen, and said, work on this activity together and keep using that pen.
2
SPEAKER_00
So then we had all these pieces of student work.
2
SPEAKER_00
And what we saw was, they just worked on separate parts of the paper.
2
SPEAKER_00
So there s a little like red pen section and a green pen section.
2
SPEAKER_00
And they didn t do that well on it, even though it was a problem that middle or high school kids could do, but it was like a problem solving kind of problem.
2
SPEAKER_00
And then we gave them the same one to do at the end, gave them the same colors, and actually they had learned to collaborate.
2
SPEAKER_00
And not only were they collaborating the second time around, but that boosted their achievement.
2
SPEAKER_00
And the ones who collaborated did better on the problem.
2
SPEAKER_00
collaboration is important, having people and what was so eye opening for these undergrads, and they talked about it in lovely ways was, I learned to value other people s thinking on a problem.
2
SPEAKER_00
And I learned to value that other people saw it in different ways.
2
SPEAKER_00
And
2
SPEAKER_00
It was quite a big experience for them that they came out thinking, you know, I can do maths with other people.
2
SPEAKER_00
People can see it differently.
2
SPEAKER_00
We can build on each other s ways of thinking.
2
SPEAKER_01
I got a chance to, I don t know if you know who Daniel Kahneman is, got a chance to interact with him.
2
SPEAKER_01
And like the first, cause he had a few, but one famous collaboration throughout his life with Tversky.
2
SPEAKER_01
Just like, you know, he hasn t met me before, in person, but just the number of questions he was asking, just the curiosity.
2
SPEAKER_01
So I think one of the skills, the collaboration itself is a skill.
2
SPEAKER_01
And I remember my experience with him was like, okay, I get why you re so good at collaboration.
2
SPEAKER_01
Because he was just extremely good at listening, and genuine,
2
SPEAKER_01
curiosity about how the other person thinks about the world, sees the world.
2
SPEAKER_01
And then together, he pulled me in, in that particular case, he doesn t know in particular like that much about autonomous vehicles, but he kept like asking all of these questions.
2
SPEAKER_01
And then like 10 minutes in, we re together trying to solve the problem of autonomous driving.
2
SPEAKER_01
And that s really fulfilling, that s really enriching, but it also, in that moment, made me realize it s kind of a skill.
2
SPEAKER_01
You have to put your ego aside, put your view of the world aside, and try to learn how the other person sees it.
2
SPEAKER_00
And the other thing you have to put aside is this social comparison thinking.
2
SPEAKER_00
If you are sitting there thinking, wow, that was an amazing idea, he s so much better than I am, that s really gonna stop you taking on the value of that idea.
2
SPEAKER_00
So there s a lot of that going on between these Stanford students when they came.
2
SPEAKER_00
And trying to help them let go of that.
2
SPEAKER_01
One of the things I ve discovered, just because being a little bit more in the public eye, how rewarding it is to celebrate others.
2
SPEAKER_01
And how much it s going to actually pay off in the long term.
2
SPEAKER_01
So this kind of silo thinking of like, I want to prove to a small set of people around me that I m really smart.
2
SPEAKER_01
and do so by basically not celebrating how smart the other people are, that s actually maybe short term, it seems like a good strategy, but long term it s not.
2
SPEAKER_01
And I think if you practice at the student level and then at the career level, at every single stage, I think that s ultimately.
2
SPEAKER_00
I agree with you, I think that s a really good way of thinking about it.
2
SPEAKER_01
You mentioned textbooks and you didn t say it, you know, maybe textbooks isn t the perfect way to teach mathematics, but I love textbooks.
2
SPEAKER_01
They re like pretty pictures and they smell nice and they open.
2
SPEAKER_01
I mean, I talk about like physical.
2
SPEAKER_01
Some of my greatest experiences have been just like, oh, like, cause they re really well done.
2
SPEAKER_01
When we re talking about basic, like high school calculus,
2
SPEAKER_01
biology, chemistry, those are incredible.
2
SPEAKER_01
It s like Wikipedia, but with color and nice little stuff.
2
SPEAKER_00
You must have seen some good textbooks.
2
SPEAKER_00
They had pretty pictures in color.
2
SPEAKER_01
Yeah, I mean, I remember, I guess it was very, very standard, like AP calculus, AP biology, AP chemistry.
2
SPEAKER_01
I felt those were like some of the happiest days of my life in terms of learning was high school, because it was very easy, honestly.
2
SPEAKER_01
It felt hard at the time, but you re basically doing a whirlwind tour of all the science.
2
SPEAKER_01
Without having to pick, you do literature, you do Shakespeare, you do calculus, biology, physics.
3
SPEAKER_00
chemistry, what else, anatomy, physiology, computer science, without like, nobody s telling you what to do with your life, you re just doing all of those things.
3
SPEAKER_01
That s a good thing, you re right.
3
SPEAKER_00
But I remember the textbooks weren t, I mean, maybe I m romanticizing the past, but I remember they weren t, they were pretty good.
3
SPEAKER_00
But so you think, what role do you think they play still?
3
SPEAKER_00
And like in this more modern digital age, what s the best materials with which to do these kinds of educations?
3
SPEAKER_01
Well, I m intrigued that you have such a good experience with textbooks.
3
SPEAKER_01
I mean, I can remember loving some textbooks I had when I was learning and I love books.
3
SPEAKER_01
I love to pick up books and look through them.
3
SPEAKER_01
But
3
SPEAKER_01
A lot of maths textbooks are not good experiences for kids.
3
SPEAKER_01
We have a video on our website of the kids who came to our camp and one of the students says,
3
SPEAKER_01
In maths, you have to follow the textbook.
3
SPEAKER_01
The textbook s kind of like the Bible.
3
SPEAKER_01
You have to follow it.
3
SPEAKER_01
And every day, it s slightly different.
3
SPEAKER_01
Like on Monday, you do 2.3.2, and on Tuesday, you do 2.3.3, and on Wednesday.
3
SPEAKER_01
And you never go off that.
3
SPEAKER_01
That s like every single day.
3
SPEAKER_01
And that s not inspiring for a lot of the kids.
3
SPEAKER_01
So one of the things they loved about our camp was just that there were no books.
3
SPEAKER_01
Even though we gave them sheets of paper instead, they still felt more free.
3
SPEAKER_01
because they weren t just like trotting through exercises, exercises.
3
SPEAKER_00
So... Like what a textbook allows you is like, the very thing you said, they might not like the 2.3, it feels like you re making progress.
3
SPEAKER_00
And like it s a little celebrations because you do the problem and it seems really hard and you don t know how to do it.
3
SPEAKER_00
And then you,
3
SPEAKER_00
try and try and then eventually succeed.
3
SPEAKER_00
Then you make that little step and further progress.
3
SPEAKER_00
And then you get to the end of a chapter and you get to like, it s closure.
3
SPEAKER_00
You re like, all right, I got that figured out.
3
SPEAKER_00
And then you go on to the next chapter.
3
SPEAKER_01
I can see that.
3
SPEAKER_01
I mean, I think it could be in a textbook, you can have
3
SPEAKER_01
a good experience with a textbook.
3
SPEAKER_01
But what s really important is what is in that textbook?
3
SPEAKER_01
What are you doing inside it?
3
SPEAKER_01
I mean, I grew up in England, and in England, we learn maths.
3
SPEAKER_01
We don t have this separation of algebra and geometry.
3
SPEAKER_01
And I don t think any other country apart from the US has that.
3
SPEAKER_01
But I look at kids in algebra classes where they re doing algebra for a year, and I think I would have been pretty bored doing that.
3
SPEAKER_00
By the way, can we analyze your upbringing real quick?
3
SPEAKER_00
Why do British folks call mathematics maths?
3
SPEAKER_00
Why is it the plural?
3
SPEAKER_00
Is it because of everything you re saying, or is it a bunch of sub-disciplines?
3
SPEAKER_01
Yeah, I mean, mathematics is supposed to be the different maths that you look at, whether you think of that as topics like geometry and probability, or I think of it as
3
SPEAKER_01
maths is just multidimensional in lots of ways.
3
SPEAKER_01
But that s why it was called mathematics.
3
SPEAKER_01
And then it was shortened to maths.
3
SPEAKER_01
And then, for some reason, it was just math in the US.
3
SPEAKER_01
But to me, math has that more singular feel to it.
3
SPEAKER_01
And there s an expression here, which is do the math, which basically means do a calculation.
3
SPEAKER_01
That s what people mean by do the math.
3
SPEAKER_01
So I don t like that expression because math could be anything.
3
SPEAKER_01
It doesn t have to be calculation.
3
SPEAKER_01
So yeah, I like maths because it has more of that broad feel to it.
3
SPEAKER_00
I love that.
3
SPEAKER_00
Maths kind of emphasizes the multidimensional, like a variety of different disciplines, different approaches, yeah.
3
SPEAKER_00
Okay.
3
SPEAKER_00
But outside of the textbook, what do you see broadly being used?
3
SPEAKER_00
You mentioned Sebastian Thrun and MOOCs, online education.
3
SPEAKER_00
Do you think that s an effective set?
3
SPEAKER_01
It can be.
3
SPEAKER_01
I mean, online.
3
SPEAKER_01
having great teachers online obviously extends those teachers to many more people.
3
SPEAKER_01
And that s a wonderful thing.
3
SPEAKER_01
I have quite a few online courses myself.
3
SPEAKER_01
I got the bug working with Sebastian when he had released his first MOOC.
3
SPEAKER_01
And I thought, maybe I could do one in maths education.
3
SPEAKER_01
And I didn t know if anybody would take it.
3
SPEAKER_01
I remember releasing it that first summer and it was a free online class and 30,000 maths teachers took it that first summer and they were all talking about it with each other and sharing it and it was like took off.
3
SPEAKER_01
In fact, it was that MOOC that caught
3
SPEAKER_01
got me to create You Cubed with Kathy Williams, who s the co founder.
3
SPEAKER_01
Because people took the MOOC and then they said, Okay, what now?
3
SPEAKER_01
I finished what, what can I have next?
3
SPEAKER_01
And so that was where we made our website.
3
SPEAKER_01
But so yeah, I think online education can be great.
3
SPEAKER_01
I do think a lot of the MOOCs don t have great pedagogy.
3
SPEAKER_01
They re just a talking head.
3
SPEAKER_01
And
3
SPEAKER_01
you can actually engage people in more active ways even in online learning.
3
SPEAKER_01
So I learned from the Udacity principle when I was working at Udacity never to talk more than like five minutes.
3
SPEAKER_01
And then to ask people to do something.
3
SPEAKER_01
So that s the sort of pedagogy of the online classes I have.
3
SPEAKER_01
There s a little bit of presenting something and then people do something and there s a little bit more.
3
SPEAKER_01
Because I think if you have a half hour video, you just,
3
SPEAKER_01
I switch off and start doing other things.
3
SPEAKER_00
So the way you d actually did it is like five, 10 minute, like bit of teaching with some visual stuff perhaps.
3
SPEAKER_00
And then there s like a quiz almost.
3
SPEAKER_01
Then you answer a question, yeah.
3
SPEAKER_00
Yeah, that s really effective.
3
SPEAKER_00
You mentioned YouCubed.
3
SPEAKER_00
So what s the mission?
3
SPEAKER_00
What s the goal?
3
SPEAKER_00
You mentioned how it started, but what s, yeah, where are you at now?
3
SPEAKER_00
And what do you, what s your dream with it?
3
SPEAKER_00
Or what are the kind of things that people should go and check out on there?
3
SPEAKER_01
Yeah, we started YouCubed, I guess it was about five years ago now.
3
SPEAKER_01
And we ve had over 52 million visitors to the site.
3
SPEAKER_01
So I m very happy about that.
3
SPEAKER_01
And our goal is to share good ideas for teaching with
3
SPEAKER_01
teachers, students, parents in maths.
3
SPEAKER_01
And to help, we have a sort of sub goal of a raising maths anxiety, that s important to us, but also to share maths as this beautiful creative subject.
3
SPEAKER_01
And it s been really great.
3
SPEAKER_01
We have lessons on the site.
3
SPEAKER_01
But one of the reasons I thought this was needed is there s a lot of knowledge in the academy about how to teach maths well, loads and loads of
3
SPEAKER_01
research and journals and lots of things written up.
3
SPEAKER_01
But teachers don t read it.
3
SPEAKER_01
They don t have access to it.
3
SPEAKER_01
They re often behind pay walls.
3
SPEAKER_01
They re written in really inaccessible ways, so people wouldn t want to read them or understand them.
3
SPEAKER_01
So this, I see, is a big problem.
3
SPEAKER_01
You have this whole industry of people finding out how to teach well, not sharing it with the people who are teaching.
3
SPEAKER_01
So that s where we made YouCubed.
3
SPEAKER_01
And instead of just putting articles up saying, here s some things to read about how to teach well, we translated what was coming from research into things that teacher could use.
3
SPEAKER_01
So lessons, there were videos to show kids, and there were tips for parents, there were all sorts of things on the site.
3
SPEAKER_01
And it s been amazing.
3
SPEAKER_01
As we had, we took inspiration from the week of code, which
3
SPEAKER_01
got teachers to focus on coding for a week.
3
SPEAKER_01
And we have this thing called the Week of Inspirational Maths.
3
SPEAKER_01
And we say, just try it for a week.
3
SPEAKER_01
Just give us one week and try it and see what happens.
3
SPEAKER_01
And so it s been downloaded millions of times.
3
SPEAKER_01
Teachers use it every year.
3
SPEAKER_01
They start the school year with it.
3
SPEAKER_01
And what they tell us is, it was amazing.
3
SPEAKER_01
The kids  lights were on.
3
SPEAKER_01
They were excited.
3
SPEAKER_01
They loved it.
3
SPEAKER_01
And then the week finished and I opened my textbooks and the lights went out and they were not interested.
3
SPEAKER_00
Yeah.
3
SPEAKER_00
But getting that first inspiration is still powerful.
3
SPEAKER_01
It is.
3
SPEAKER_01
I wish, I mean, what I would love is if we could actually extend that for the whole year.
3
SPEAKER_01
We re a small team at Stanford and we re trying to keep up with great things to put on the site.
3
SPEAKER_01
we haven t the capacity to produce these creative visual math tasks for every year group for every day, but I would love to do that.
3
SPEAKER_00
How difficult is it to do?
3
SPEAKER_00
I mean, it s to come up with visual formulations of these big important topics you need to think about in a way that, you know, that you could teach.
3
SPEAKER_01
I mean, it we can do it.
3
SPEAKER_01
We actually we went from the week of inspirational maths, and we made k eight maths books with exactly that big ideas, rich activities, visuals, we just finished the last one, we ve been doing it for five years, and it s been exhausting.
3
SPEAKER_01
And we just finished.
3
SPEAKER_01
So now there s a whole k eight set of books.
3
SPEAKER_01
And
3
SPEAKER_01
And they re organized in that way.
3
SPEAKER_01
These are the big ideas.
3
SPEAKER_01
Here are rich, deep activities.
3
SPEAKER_01
They re not, though, what you can do every day for a year.
3
SPEAKER_01
So some teachers use them as a kind of supplement to their boring textbook.
3
SPEAKER_01
And some people have said, okay, this is the year.
3
SPEAKER_01
This book tells us what the year is.
3
SPEAKER_01
And then we ll supplement these big activities with
3
SPEAKER_01
So they re being used and teachers really like them and are really happy about them.
3
SPEAKER_01
I just always want more.
3
SPEAKER_01
And I guess one of the things I would like for UCubed, one of my personal goals is that every teacher of maths knows about UCubed.
3
SPEAKER_01
At the moment, lots of teachers who come to us are really happy they found it, but there s a lot of other teachers who don t know that it exists.
3
SPEAKER_00
I hope this helps.
3
SPEAKER_01
Yeah.
3
SPEAKER_00
From a student perspective and not in the classroom but at home studying, is there some advice you can give on how to best study mathematics?
3
SPEAKER_00
So what s the role of the student outside the classroom?
3
SPEAKER_01
Yeah, I think one thing we know is a lot of people when they review material, whether it s maths or anything else, don t do it in the best way.
3
SPEAKER_01
I think a problem a lot of people have is they read through maybe a teacher s explanation or a way of doing maths.
3
SPEAKER_01
And it makes sense.
3
SPEAKER_01
And they think, oh, yeah, I ve got that and they move on.
3
SPEAKER_01
But then it s not until you come to try and work on something and do a problem that you actually realize you didn t really understand it just seemed to make sense.
3
SPEAKER_01
So I would say, this is also something that neuroscientists talk about, to keep giving yourself questions is a really good way to study, rather than looking through lots of material.
3
SPEAKER_01
It s always like giving yourself lots of tests.
3
SPEAKER_01
is a good way to actually deeply understand things and know what you do when you don t understand.
3
SPEAKER_00
So would the questions be in the form of the material you re reviewing is the answer to that question?
3
SPEAKER_00
Or is it almost like beyond, it s the polygon thing that you mentioned from a square.
3
SPEAKER_00
Is it almost like, I wonder what is the bigger picture?
3
SPEAKER_00
I was kind of asking of like, how is this extended and so on.
3
SPEAKER_01
Yeah.
3
SPEAKER_01
That would be great.
3
SPEAKER_01
And it s a similar, I mean, a question I get asked a lot is about homework, what is a good thing for kids to do for homework?
3
SPEAKER_01
And one of the recommendations I give is to not have kids just do lots of questions for homework, but to actually ask them to reflect on what they ve learned.
3
SPEAKER_01
Like, what was the big idea you learned today?
3
SPEAKER_01
Or what did you find difficult?
3
SPEAKER_01
What did you struggle with?
3
SPEAKER_01
What was something that was exciting?
3
SPEAKER_01
Then kids go home and they have to kind of reflect in a deeper way.
3
SPEAKER_01
A lot of times, I don t know if you had this experience as a math student, lots of people do, kids are going through math questions, they re successful, they get them right, but they don t even really know what they re about.
3
SPEAKER_01
And a lot of kids go through many years of math like that, doing lots of questions, but really knowing what even,
3
SPEAKER_01
the topic is or what it s about, what it s important for.
3
SPEAKER_01
So having students go back and think at the end of the day, what was the big idea from this maths lesson?
3
SPEAKER_01
Why is it important?
3
SPEAKER_01
Where would I find that in real life?
3
SPEAKER_01
Those are really good questions for kids to be thinking about.
3
SPEAKER_00
Probably for everybody to be thinking about.
3
SPEAKER_00
I think most of us go through life never asking like the bigger question, almost like, you know, those like layers of why questions that kids ask when they re very young.
3
SPEAKER_01
Yeah.
3
SPEAKER_00
We need to keep doing that.
3
SPEAKER_00
We do.
3
SPEAKER_00
Like what that s the, you know, whatever the term is, you call first principles thinking some people call it that, which is like,
3
SPEAKER_00
Why are we doing it this way?
3
SPEAKER_00
So one nice thing is to do that, because there s usually a good answer.
3
SPEAKER_00
The reason we did it this way is because it works for this reason.
3
SPEAKER_00
But then if you want to do something totally novel, you ll say, well, we ve been doing it this way because of historical reasons.
3
SPEAKER_00
But really, this is not the best way to do it.
3
SPEAKER_00
There might be other ways.
3
SPEAKER_00
And that s how invention happens.
3
SPEAKER_00
And then you get, that s really useful in every aspect of life, like choosing your career, choosing your, I don t know, where you live, who your romantic partner is, everything.
3
SPEAKER_00
And I think it probably starts doing that in math class.
3
SPEAKER_00
Yeah.
3
SPEAKER_01
That would be good if we started doing that.
3
SPEAKER_00
I mean, I wonder,
3
SPEAKER_00
I probably didn t do very much of that for most of my education, asking why, except for later, much later in the subjects on like grad school when you re doing research on them.
3
SPEAKER_00
When you re first tasked with doing something novel using this,
3
SPEAKER_00
or solving a problem really outside the classroom.
3
SPEAKER_00
They have to publish on it.
3
SPEAKER_00
It s the first time you think, wait, why are these things interesting, useful?
3
SPEAKER_00
Which are the things that are useful?
3
SPEAKER_00
And yeah, I guess that would be nice if we did that much earlier, the quest of invention.
3
SPEAKER_01
Yeah, yeah.
3
SPEAKER_01
I mean, one of the sad pieces of research data I think about is the questions kids ask,
3
SPEAKER_01
in school goes down in a linear progression.
3
SPEAKER_01
In the early years, you can t stop kids asking those questions, but they learn not to ask the questions.
3
SPEAKER_00
I think you told somewhere about an early memory you had in your own education where you asked the question, or maybe that was an example you gave, but it was shut down.
3
SPEAKER_01
Oh, yeah.
3
SPEAKER_01
You ve listened to something I said, yeah.
3
SPEAKER_00
I don t remember where it was, but it caught me.
3
SPEAKER_01
Yeah, I remember it really vividly.
3
SPEAKER_00
Well, can you tell the memory?
3
SPEAKER_01
Yeah, I was.
3
SPEAKER_01
It s funny.
3
SPEAKER_01
I can remember it must have really impacted me in that moment.
3
SPEAKER_01
Because you know how there s lots of hours of school you don t remember at all.
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But anyway, I can remember where I was sitting and everything.
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I was in high school maths class, although they don t call it that in England.
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And I
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The teacher said, and it was like the first class of this teacher s class, and he said, ask if you have any questions.
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So at one point I put my hand up and I said, I have a question.
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And he said something like, that s your question?
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And I was like, oh, okay, I m not asking any more questions in this class.
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And then it hit hard in a way where you didn t wanna, the lesson you learned from that is I m not gonna ask any more questions.
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Yeah, that was absolutely, that s the last question I m asking.
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That was, yeah, he was the chair of the maths department, I remember that really well.
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So maybe because of that experience, one of the things we encourage when we teach kids is asking questions, and we value it when they ask questions, and we put them up on walls and celebrate.
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It s funny because
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I wish there was a feedback signal because he probably, to put a positive spin on it, he probably didn t realize the negative impact he s had in that moment, right?
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If he only knew.
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See, this is probably when you re more mature in grad school.
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I had an amazing professor named Alisha Kafandeh in computer science.
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And he would get, he encouraged questions, but then he would tell everybody how dumb their questions are.
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But it was, it was done, I guess if you show, if you say it with love and respect behind it, then it s more like a friendly, humorous encouragement for more questions.
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It s an art, right?
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To do it right.
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And then you have to time it right because that s probably, that kind of humor is probably better for when you re in grad school versus when you re in the early education.
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Right.
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Well, and I guess kids, or young people get whether somebody is doing it to be funny, or, you know, has this, I mean, this is why it teaches so hard, even your tone can be impactful.
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It s so sad because for that particular human, the teacher, you just had a bad day, and one statement can have a profound negative impact.
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But I know, sadly, that there s a lot of math teachers who have that kind of approach, and I think they re suffering from the fact that they think people are math people, not math people, and that comes across in their teaching.
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But on the flip side, one positive statement, keep them going.
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That s right.
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That is the flip side of that.
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And I myself had like one teacher who was really amazing.
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for me in maths and she kept me in the subject.
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Who was she?
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Her name was Mrs. Marshall.
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She was my A-level maths teacher.
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What does that mean?
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In England, you do lots of subjects until you re 16 and then you choose
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like three or four subjects.
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So I had chosen maths and you go to higher levels, probably equivalent more to a master s degree in the US because you re more specialized.
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But anyways, she was my teacher.
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And for the first time in my whole career in maths, she would give us problems and tell us to talk about them with each other.
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And so here I was sitting there like 17 talking with friends about how to solve a math problem.
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And that was it.
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That was the change that she made.
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But it was profound for me.
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Because like those calculus students, I started to hear other people s ways of thinking and seeing it.
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We would talk together and come up with solutions.
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And I was like, that was it.
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That changed maths for me.
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So it wasn t some kind of personal interaction with her.
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It was more like she was the catalyst for that collaborative experience.
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I mean, yeah, the many ways teachers can inspire kids.
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I mean, sometimes it s a personal message, but it can be your teaching approach that changes maths for kids.
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You know, Cal Newport, he wrote a book called Deep Work.
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And he s a mathematician, a theoretical computer scientist.
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And he talks about the kind of the focus required to do that kind of work.
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Is there something you can comment on?
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We live in a world full of distractions.
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That seems like one of the elements that makes studying, and especially the studying of subjects that require thinking like math does, difficult.
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Is there something from a student perspective, from a teacher perspective that encourages deep work that you can comment on?
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Giving kids really inspiring, deep problems, and we have some on our website, is a really important experience for them.
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Even if they only do it occasionally, but it s really important.
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They actually realize, I give a problem out often when I m working with teachers, and I say to them, all right, I m gonna check in with you after an hour.
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And they re like, an hour, they think it s shocking.
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And then they work on this problem.
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And after an hour, I say, Okay, how are we doing?
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An hour has gone by.
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How is this possible?
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And so everybody needs those, like rich, deep problems.
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Most kids go through their whole maths experience of however many years never once working on a problem in that kind of deep way.
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So I, the undergrad class I teach at Stanford, we do that we work on these deep problems, every session, and the students come away going, Okay, I never want to go back to that maths relationship I had where it was just all about quick answers.
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I, I just don t want to go back to that.
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And so
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We can all, all teachers can incorporate those problems in their classrooms.
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Maybe they don t do them every day, but they at least give kids some experience of being able to work slowly and deeply and to go to deeper places and not be told they ve got five minutes to finish 20 questions.
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Yeah.
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Well, part of it is also just the, um,
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the exercise of sitting there maintaining focus for prolonged periods of time.
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That s not often, I mean, that s a skill.
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It s a skill that also could be discouraging.
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If you don t practice it,
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Just sitting down for 10 minutes straight and maintaining deep focus can be exceptionally challenging.
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Like if you re really thinking about a problem.
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I think it s really important to realize that that s a skill that you can, just like a muscle, you can build, you can start with 5 minutes and go to 10 minutes to 30 and to an hour.
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And to be successful, I think in certain subjects, like mathematics, you wanna be able to develop that skill.
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Otherwise, you re not going to get to the really rewarding experience of solving these problems.
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Definitely.
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There was a survey done of kids in school where they were asked, how long will you work on a maths problem before you give up and decide it s not possible to solve it?
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And the result on average across the kids was two minutes.
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That s a bad sign, but that was a powerful sign that they need to learn to not give up so quickly.
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Yeah.
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We mentioned offline, because we ve been talking so much about visualization, Grant Sanderson, 3Luan Brown.
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So, he s inspired millions of people with exactly the kind of way of thinking that you ve been talking about.
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Yeah, I love his...
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converting sort of mathematical concepts into visual, like visually representing them, exploring them in ways that help you illuminate the concepts.
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What do you think is the role of that?
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So he uses mostly programmatic visualizations, so it s the thing I mentioned where there s like animations created by writing computer programs.
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Like what do you think, how scalable is that approach?
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But in general, what do you think about his approach?
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I think it s amazing.
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I should work with him.
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I can share some of our visuals and he can make them in that amazing way.
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So part of his storytelling, part of it is like,
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It s creating the visuals and then weaving a story with those visuals that kind of builds.
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There s also, I mean, there s also drama in it.
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You start with a small example and then you kind of, all of a sudden there s a surprise.
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And it really, I mean, it makes you fall in love with the concept.
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He does talk about that.
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his sense is like some of the stuff, he doesn t feel like he s teaching like the core curriculum, which is something, you know, he sees himself as an inspirational figure.
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But because I think it s too difficult to kind of convert all of the curriculum into those elements.
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And probably you don t need to.
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I mean, you, if people get to experience mathematical ideas in the way that he shares them,
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that will change them and it will change the way they think and maybe they could go on to take some other mathematical idea and make it that beautiful.
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Well, he does that, there s a, he created a library called Manum and he open sourced it.
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And that library is the, people should check it out.
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It s written in Python and uses some of those same elements.
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Like it allows you to animate equations and animate little shapes, like people that, you know, he has a very distinct style in his videos.
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And what that resulted in, even though from a software engineer perspective, the code he released is not like,
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super well-documented or perfect, but him releasing that, now there s all of these people educating it.
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And the cool, to me personally, the coolest thing is to see like people they re not, you know, don t have like a million subscribers or something.
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They have just a few views in the video, but it just seems like the process of them creating a video where they teach
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is transformative to them from a student perspective.
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It s the old Feynman thing, the best way to learn is to teach.
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And then him releasing that into the wild, it shows that impact.
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Yeah, absolutely.
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I think just giving people that idea that you can do that with maths and other subjects, there s bound to be people all around who can create more, which is cool.
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Yeah, definitely.
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So I recommend people do like JavaScript or Python.
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You can build like visualizations of most concepts in high school math.
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You can do a lot of kinds of visualizations and doing that yourself.
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Plus, if you do that yourself, people will really love it.
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People actually...
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People love visualizations of matter.
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I mean, it s something in us that loves patterns, loves figuring out difficult things, and the patterns that are unexpected in some way.
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Have you ever noticed that hotels are always filled with patterns?
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I was just noticing it at the hotel I m in now.
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All of their carpets are pattern carpets, and then they have patterns on the walls.
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Humans love the symmetry in patterns, the breaking of symmetry in patterns.
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And it s funny that we don t see mathematics as somehow intricately connected to that, but it is.
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I mean, that s one of the perspectives I love students to take is to be a pattern seeker.
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In everything.
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Yeah, certainly in all of maths.
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I mean, you can think of all of maths as a kind of subject of patterns and not just visual patterns.
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But, you know, when you think about multiplying by five, and the fact you can, you know, if you if you re multiplying 18 times five, you can instead think of nine times 10.
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That s a pattern that always works in mathematics.
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You can have a number and double the number.
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And so, yeah, I just think there are patterns everywhere.
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And if kids are thinking their role is to see patterns and find patterns, it s really exciting.
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What do you think about like MIT OpenCourseWare and the release of lectures by universities?
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I think it s good.
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I think it s good.
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I think that is what started
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The MOOC I did was using that platform.
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So you ultimately think like the Udacity models is a little bit more effective than just a plain two-hour lecture?
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I think there s definitely, you can bring in good pedagogy into online learning.
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And I think the idea of putting things online so that people all over the world can access them is great.
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I don t think the initial excitement around MOOCs, sort of democratizing education and making it more equal,
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came about because they found that the people taking MOOCs tended to be the more privileged people.
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So that was, I think there s still something to be found in that.
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There s still more to be done to help that online learning reach those principles.
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But definitely I think it s a good invention.
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And I have an online class that s for kids, that s a little free class that gives them.
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It s called How to Learn Math.
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It shows maths as this visual creative subject and it shares mindset and some brain science and kids who take it do better in maths class.
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We ve studied it with like randomized controlled trials and given it to middle school kids and other middle school kids who don t take it but are taught by the same teachers so their teachers are the same.
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And the kids who take the online class end up 68% more engaged in their maths class.
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do better at the end of the year.
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So that s a little six session, 15 minute class, and it changes kids maths relationship.
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So it is true that we can do that with some words that aren t, you know, not, it s not a huge change to the education system.
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Do you have advice for young people?
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We ve been talking about mathematics quite a bit, but in terms of their journey through education, through their career choices, through life, maybe middle school, high school, undergrad students, how to live a life they can be proud of?
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I think if I were to give advice to people, especially young people, my advice would be to
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always, it sounds really corny, but always believe in yourself and know that you can achieve.
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Because although that sounds like obvious, of course, we want kids to know that they can achieve things.
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I know that
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Millions of kids who are in the school system have been given the message they cannot do things.
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And adults too.
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They have the idea, oh, I did okay in this.
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I went into this job because those other things I could never have done okay in.
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So actually when they hear, hey, maybe you could do those other things,
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Even adults think, you know, maybe I can and they go back and they encounter this knowledge and they relearn things and they change careers and amazing things happen.
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So for me, I think that message is really important.
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You can learn anything.
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scientists try and find a limit.
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They re always trying to find a limit, like how much can you really learn?
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What s the limit to how much you can learn?
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And they always come away not being able to find it.
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People can just go further and further and further.
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And that is true of people born with brain, you know, areas of their brain that aren t functioning well, that have what we call special needs.
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Some of those people also go on to develop and do amazing things.
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So
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I think that really experiencing that, knowing that, not just saying it, but knowing it deeply, you can learn anything, is something I wish all people would have.
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actually also applies when you ve achieved some level of success too.
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What I find, like in my life with people that love me, when you achieve success, they keep celebrating your success and they want you to keep doing the thing that you were successful at as opposed to believing in that you can do something else, something big, whatever your heart says to do.
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And one of the things that I realized the value of this,
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you know, quite recently, which is sad to say, is how important it is to seek out, when you re younger, to seek out mentors, to seek out the people, like surround yourself with people that will believe in you.
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It s like a little bit is on you.
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It s like you don t get that sometimes if you go to grad school, you think you kind of land on a mentor, maybe you pick a mentor based on the topic they re interested in, but the reality is the people you surround yourself with,
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They re going to define your life trajectory.
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So select people that... That s really true.
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And get away from people who don t believe in you.
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Sometimes parents can be that.
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They can love you deeply, but they set... It s the math thing we mentioned.
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They might set certain constraints on the beliefs that you have.
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So in that, if you re interested in mathematics and your parents are not that interested in it, don t listen to your parents on that one direction.
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Exactly, yeah.
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And if people tell you you can t do things, you have to hear from other people who believe in you.
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I think you re absolutely right about that.
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So sad the number of people who ve had those negative messages from parents.
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In my Limitless Mind book, I interviewed quite a few people who d been told they couldn t do math, sometimes by parents, sometimes by teachers.
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And fortunately, they had got other ideas at some point in their life and realized there was this whole world of mathematical thinking that was open to them.
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So it s really important that people do connect with people who believe in them, however hard that might be to find those people.
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What do you hope the education system, education in general, looks like 10, 20, 50, 100 years from now?
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Are you optimistic about this future?
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Yeah, I definitely have hope.
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There is, change can happen in the education system.
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In recent years, it s been microscopically slow.
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But
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I do actually see change happening.
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Like we were talking earlier that data science is now a course you can take in high school instead of algebra two.
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And that s pretty amazing because that content was set out in 1892 and hasn t changed since then.
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And so now we re actually seeing a change in the content of high school.
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So I m amazed that that s happening and very happy it s happening.
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But so change is very slow in education usually, but
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When you look ahead and think about all that we know and all that we can offer kids in terms of technology, you ve got to think that 100 years from now, education will be totally different to the way it is now.
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Maybe we won t have subject boundaries anymore.
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because those don t really make much sense.
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And it s interesting to think how certain tools like programming, maybe they ll be deeply integrated in everything we do.
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Yeah, you would think that all kids are growing up learning to program and create.
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So I just think, I mean, the system of schooling we have now, people call it a factory model.
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It s not designed to inspire creativity and
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I feel like that will also change.
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People might look back on these days and think they were hilarious, but maybe in the future, kids will be doing their own programming and they ll be able to learn things and find out things and create things even as they re learning.
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Maybe the individual subjects boundaries will go.
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Data science itself coming into the education system kind of illustrates that because people realize it doesn t really fit inside any of the subjects.
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So what do we do with it?
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Where does it go?
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And who teaches it?
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So it s already raising those kind of questions and
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questioning how we have these different subject boundaries.
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So you ve seen data science be integrated into the curriculum?
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Yes, it s happening across the United States as we speak.
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I wonder how that got initiated.
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How does change happen in the education system?
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Is it just a few revolutionary leaders?
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It does, I think so, I think so.
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It s been an interesting journey seeing data science take off actually.
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There was a,
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course that was developed in 2014, by some people who thought data science was a good idea for high schoolers.
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And then after some kids took the course and nothing bad happened to them, they went to college and people started to accept it more.
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And then this was a big piece of the change in California, the UC system communicated, they sent out an email last year to 50,000 high schools saying we now accept data science, kids can take it instead of algebra two, that s a perfectly legitimate college pathway.
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So that was like a big green light for a lot of schools who were like,
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wondering about whether they could teach it.
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So I think it happens in small spaces and expands.
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SPEAKER_00
So now- It goes viral.
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SPEAKER_01
Yeah.
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In this modern age.
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Then it goes viral.
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California s ahead, I think, in creating courses and having kids go through it.
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But it s suddenly, when I last looked, there were 12 states that were allowing data science as a high school course.
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And I think by next year, that will have doubled or more.
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So
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A change is happening.
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As I said, I think mathematics is truly a beautiful subject and you having an impact on millions of people s lives by educating them, by inspiring teachers to educate in the ways that you ve talked about, in multidimensional ways, in visual ways, I think is incredible.
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So you re spreading beauty into the world.
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So I really, really appreciate that you spent your valuable time with me today.
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Thank you for talking.
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SPEAKER_01
Thank you, it was really good to talk to you.
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SPEAKER_00
Thanks for listening to this conversation with Joe Bowler.
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To support this podcast, please check out our sponsors in the description.
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SPEAKER_00
And now let me leave you with some words from Albert Einstein.
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Pure mathematics is the poetry of logical ideas.
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Thanks for listening and hope to see you next time.