back to indexJo Boaler: How to Learn Math | Lex Fridman Podcast #226
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The following is a conversation with Joe Bowler,
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a mathematics educator at Stanford
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and cofounder of ucubed.org
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that seeks to inspire young minds
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with the beauty of mathematics.
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To support this podcast,
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please check out our sponsors in the description.
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This is the Lex Friedman podcast
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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
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don't even think of as mathematics,
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which is beautiful creative mathematics
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where we look at maths in different ways.
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We think about different solutions to problems.
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A lot of people think of maths
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as you have one method and one answer.
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And what I love about maths
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is the multiple different ways you can see things.
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Different methods, different ways of seeing, different.
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In some cases, different solutions.
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So that is what is beautiful to me about mathematics
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that you can see and solve it in many different ways.
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And also the sad part that many people think
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that maths is just one answer and one method.
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So to you, the beauty emerges
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when you have a problem with a solution
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and you start adding other solutions.
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Simpler solutions, weirder solutions,
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more interesting, some of their visuals,
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some of their algebraic geometry,
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all that kind of stuff.
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Yeah, I always say that you can take any maths area
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and make it visual.
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And we say to teachers, give us your most dry, boring maths
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and we'll make it a visual, interesting, creative problem.
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And turns out you can do that with any area of maths.
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And I think we've given,
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it's been a great disservice to kids and others
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that it's always been numbers, lots and lots of numbers,
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numbers can be great,
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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
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or is this just something that's complementary?
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What's the kind of the full spectrum of students
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in 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
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who are more interested in visual thinking
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and like visual approaches.
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But it turns out what the neuroscience is telling us
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is that when we think about maths,
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there are two visual pathways in the brain
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and we should all be thinking about it visually.
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Some approaches have been to say,
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well, you're a visual learner,
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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,
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it's probably more important that you have a visual approach.
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So you can develop that part of your brain.
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So you were saying that there's some kind
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of interconnected aspect to it.
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So the visual connects with the non visual.
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Yeah, so this is what the neuroscience has shown us,
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that when you work on a maths problem,
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there are five different brain pathways
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and that the most high achieving people in the world
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are people who have more connections
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between these pathways.
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So if you see a maths problem with numbers,
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but you also see it visually,
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that will cause a connection to happen in your brain
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between these pathways.
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And if you maybe write about it with words,
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that would cause another connection
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or maybe you build it with something physical
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that would cause a different connection.
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And what we want for kids is,
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we call it a multi dimensional experience of maths,
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seeing it in different ways,
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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
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that part of their brain of using visualization
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for intuition building,
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because you ultimately want to understand
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the deepest secret underneath this problem.
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And for that, you have to sit into it your way there.
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And you mentioned offline that one of the ways
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you might approach a problem
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is to try to tell a story about it.
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And some of it is like legend,
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but I'm sure it's not always,
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is you have Einstein thinking about a train,
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and the speed of light,
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and that kind of intuition is useful.
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You start to imagine a physical world.
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Like 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,
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and think, is this going to be true?
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Is this going to be true?
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Right, Einstein is well known for thinking visually.
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And people talk about how he really didn't want to go anywhere
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with problems without thinking about them visually.
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But the other thing you mentioned that sparked something for me
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is thinking with intuition,
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like having intuition about math problems.
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That's another thing that's often absent in math class,
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the idea that you might think about a problem
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and use your intuition, but so important.
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And when mathematicians are interviewed,
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they will very frequently talk about the role of intuition
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in solving problems,
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but not commonly acknowledged or brought into education.
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Yeah, I mean, that's what it is.
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Like if you task yourself with building an intuition
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about a problem, that's where you start to pull in,
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like what is the pattern I'm seeing?
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And in order to understand the pattern,
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you might want to then start utilizing visualization.
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But ultimately, that's all in service of like solving the puzzle,
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like cracking it open to get the simple explanation
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of why things are the way they are,
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as opposed to, like you said,
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having a particular algorithm
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that you can then execute to solve the problem.
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Yeah, but it's hard.
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Like reasoning is really hard.
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I mean, I love to value what's hard in maths
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instead of being afraid of it.
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We know that when you struggle,
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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,
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that's probably a really good time for your brain.
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I used to work with somebody called Sebastian Thrun,
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who's a great sort of mathematician,
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you might think of him as AI person.
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And I remember in one interview I did with him,
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he talked about how they'd built robots,
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I think for the Smithsonian,
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and how they were having this trouble
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with them picking up white noise.
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And he said they had to solve it.
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They had to work out what's going on
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and how he intuitively 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
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that how you can have this intuition
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and know something works,
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it's kind of different from going through
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that long mathematical process of proving it.
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Yeah, I think probably our brains are evolved
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as like intuition machines.
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And the math of like showing it like formally
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is probably an extra thing that we're not designed for.
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You see that with Feynman and his,
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I mean, just all of these physicists,
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definitely you see starting with intuition,
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sometimes starting with an experiment,
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and then the experiment inspires intuition.
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But you can think of an experiment
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as a kind of visualization.
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Just like let's take whatever the heck we're looking at
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and draw it and draw like the pattern as it evolves
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as the thing grows for n equals one, for n equals two,
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n equals three, you start to play with it.
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And then in the modern day, which I loved doing
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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 you can do so interactively too.
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I tend to not like interactive
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because it takes 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 like 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 of those kinds of things that 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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And we love as well color coding.
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Like when you represent something mathematically,
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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,
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maybe I show the X with a certain color,
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but I also write in that color
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so you can see the relationship.
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And yeah, particularly in our work with elementary teachers,
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many of them come to our workshops
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and they're literally in tears
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when they see things making sense visually
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because they've spent their whole lives
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and not realizing you can really understand things
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with these visuals.
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It's quite powerful.
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You say that there's something about,
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there's something valuable to learning
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when the thing that you're doing is challenging, is difficult.
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So a lot of people say, you know, math is hard
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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,
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but there's something that's really key
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to being able to deal with challenging math
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and that is knowing that you can do it.
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And I think the problem in education
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is a lot of people have got this idea
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that you're either born with a math's brain or you're not.
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So when they start to struggle, they think,
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oh, I don't have that math's brain.
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And then they will literally sort of switch off in their brain
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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 can do it.
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You have to go through this struggle to get there,
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but you're able to do that.
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And so we're hampered in being able to struggle
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with these ideas we've been given about what we can do.
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That's a difficult question here.
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So there's kind of, I don't know what the right term is,
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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,
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should we make room for that?
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So how do you know the difference between this is hard
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and I don't like doing hard things
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versus my brain is wired in a way
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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.
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And people really don't appreciate
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how difficult teaching is when you're faced with,
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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
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to have this multi dimensional approach to maths.
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We've really offered it in one way,
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which is here's some numbers in a method,
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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
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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 give,
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you let kids experience it in different ways,
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maybe visually with numbers, with words,
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what happens is kids,
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there are many more kids who can access it.
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So those different brain wirings you're talking about,
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where some people are just more able to do something
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in a particular way,
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that's why we want to,
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that's one of the reasons we want to open it up.
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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
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and fell in love with math early.
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I was forced into math early
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and fell in love through force.
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Well, you fell in love without the force.
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Well, but there's something we talked about a little bit
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is there is such a value for excellence.
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It's competitive and it's also everybody kind of looks up.
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The definition of success is being in a particular class
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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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We're not, it's like it's understood,
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you're going to star on the basketball team.
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If you're gonna start on the basketball team,
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if you're going to be better than the other guys,
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the other girls on the team.
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So that coupled with the belief,
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this could be partially a communist belief, I don't know,
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but the belief that everybody is capable of being great.
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But if you're not great, that's your fault.
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And you need to work harder.
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And I remember I had a sense that probably delusional,
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but I could win a Nobel Prize.
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I don't even know what that entails.
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But I thought like my dad early on told me just off hand
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and it always stuck with me that if you can figure out
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how to build a time machine, how to travel back in time,
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it will probably give you a Nobel Prize.
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And I remember early in my life thinking,
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I'm going to invent the time machine.
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And like the tools of mathematics were in service
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of that dream of winning the Nobel Prize.
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I didn't really think in those concrete terms,
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but I just thought I could be great, that feeling.
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And then when you struggle,
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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
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that might love to hear your comments about
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is just the sheer like hours of math.
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Like the number of courses you're talking about,
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a lot of geometry, a lot more.
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I think in the American system,
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you take maybe one year of geometry.
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In high school, yeah.
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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
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and stuff like that.
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In Russia, I remember just being nailed over and over
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with John, it was just nonstop.
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And then of course, there's different perspectives
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on calculus and just the whole,
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the sense was that math is like fundamental
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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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Like 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,
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but they don't challenge adults very much with math.
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So those two things, valuing excellence
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and just a lot of math in the curriculum.
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Do you think, 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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And there is a lot of success
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of people coming through the Soviet system.
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I think something that's very different to the US
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and other countries in the world
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is that idea that excellence is important
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and you can get there if you work hard.
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In the US, there's an idea that excellence is important,
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but then kids are given the idea in many ways
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that you can either do it
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or you're one of the people who can't.
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So many students in the school system
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think they're one of the kids who can't.
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So there's no point in trying hard
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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
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in the Soviet Union, that idea is really different.
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Now, the downside of that idea
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that anybody can get there if you work hard
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is that thought that if you're not getting there,
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And I would add something into that.
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I would say that anybody can get there,
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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
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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,
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if maths was interesting and open and creative
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and multi dimensional, I would be all for it.
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We actually run summer camps at Stanford
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where we invite kids in and we give them this maths
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And in our camp classrooms, they were three hours long.
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And when we were planning, the teachers were like three hours.
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Are we gonna 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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So yeah, if maths was more like that,
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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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Is there, could you comment on what age is like
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the most important when people quit math
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or give up on themselves or on math in general?
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And perhaps that age or something earlier
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is really an important moment for them to discover,
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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
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and maths around 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
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just because they're allowed to explore
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and think in good ways in the early grades
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of elementary school.
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But fifth grade teachers are often like,
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okay, we're gonna prepare you now for middle school
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and we're gonna give you grades and lots of tests.
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And that's when kids start to feel really badly
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And so middle school years,
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we, 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,
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yes, I'm gonna keep going with STEM subjects.
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Or no, I'm not, that this isn't for me.
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So, I mean, all years are important.
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And in all years, you can kind of switch kids
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and get them on a different pathway.
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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,
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but do you think teachers should almost do like one on one,
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you know, little Johnny, I believe in you kind of thing?
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Like that energy of like.
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Turns out it's really important.
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There's a study that was done,
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it was actually done in high school English classrooms
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where all kids wrote an essay for their teacher.
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And this was done as an experiment.
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Half of the kids got feedback from their teacher,
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diagnostic feedback, which is great.
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But for half of the kids, it said an extra sentence
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at the bottom that the researchers had put on.
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And the kids who read that extra sentence
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did significantly better in English a whole year later.
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The only change was this one sentence.
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What did the sentence say?
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So what did the sentence say?
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The sentence said, I'm giving you this feedback
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because I believe in you.
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And the kids who read that did better a year later.
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So when I share this with teachers,
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I say, you know, I'm not suggesting you put on
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the bottom of all kids work and giving this feedback.
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So I believe in you.
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One of the teachers said to me, we don't put it on a stamp.
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I said, no, don't put it on a stamp.
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It's, but your words are really important.
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And kids are sitting in classrooms all the time
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thinking, what does my teacher think of me?
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Does my teacher think I can do this?
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So it turns out it is really important to be saying to kids,
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I know you can do this.
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And those messages are not given enough by teachers.
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And really believe it.
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You can't just say it.
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You have to believe it.
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I sometimes, cause it's like,
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it's such a funny dance cause I'm such a perfectionist.
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I'm extremely self critical.
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And I have one of the students come up to me
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and it's clear to me that they're not even close to good.
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And it's tempting for me to be like,
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to sort of give up on them mentally.
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But the reality is like,
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if you look at many great people throughout history,
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they sucked at some point.
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And some of the greatest took nonlinear paths
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to where they sucked for long into later life.
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And so always kind of believing
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that this person can be great.
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You have to communicate that
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plus the fact that they have to work hard.
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Silicon Valley, where I live,
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is filled with people who are dropouts at school
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or who had special needs, who didn't succeed.
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It's very interesting that have gone on to do amazing work
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I mean, I do think our school system is set up
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to value good memorizers who can reproduce
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what a teacher is showing them
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and push away those creative deep thinkers.
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Often slower thinkers, they think slowly and deeply.
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And they often get the idea early on
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that they can't be good at maths or other subjects.
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So yeah, I think many of those subject people
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are the ones who go on and do amazing things.
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So there's a guy named Eric Weinstein.
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I know many mathematicians like this,
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but he talks a lot about having a nonstandard way
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I mean, a lot of great mathematicians,
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a lot of great physicists are like that.
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And he felt like he became quickly,
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he got his PhD at Harvard,
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he became quickly an outcast of the system.
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Like the education, especially early education system
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Is there a ways for an education system
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to support people like that?
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Is it this kind of multi dimensional learning
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that you're mentioning?
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Absolutely, absolutely.
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I mean, I think education system still uses an approach
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that was in classrooms hundreds of years ago.
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The textbooks have a lot to answer for
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in producing this very uninspiring mathematics.
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But yeah, if you open up the subject
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and have people see and solve it in different ways
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and value those different ways,
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somebody I appreciated a lot is a mathematician
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called Marion Mizakani, I don't know if you've heard of her.
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She won the Fields Medal, she was from Iran.
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First woman in the world to win the Fields Medal
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in mathematics, she died when she was 40,
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she was at Stanford.
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But her work was entirely visual.
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And she talked about how her daughter thought
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she was an artist because she was always visualizing.
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And I attended, she asked me to chair the PhD defense
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for one of her students.
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And I went to the defense in the math department
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and it was so interesting because this young woman
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spent like two hours sharing her work,
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all of it was visual.
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In fact, I don't think I saw any numbers at all.
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And I remember that day thinking, wow,
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I could have brought her like 13 year old
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into this PhD defense, they would not recognize this as maths.
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But when Marion Mizakani won the Fields Medal,
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all these other mathematicians were saying
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that her work had connected
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all these previously unconnected areas of maths.
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And so, but when she was, she also shared
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that when she was in school, when she was about 13,
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she was told that she couldn't do maths.
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She was told that by her teacher.
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And this is Iran, and she grew up in Iran.
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So I love that, to be told you can't be good at maths
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and then go on and win the Fields Medal is cool.
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I've been told by a lot of people in my life
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that I can't do something.
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I'm very definitely nonstandard.
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But all it takes, that's why people talk about
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like the one teacher that changed everything.
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All it takes is one teacher.
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That's the power of that.
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So that should be inspiring to teachers.
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You as a single person, given the education system,
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given the incentives, you have the power
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to truly change lives.
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In like 20 years from now,
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I feel as a medalist will walk up to you and say thank you.
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So you did that for me.
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And I share that with teachers
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that even in this broken system
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of what they have to do for districts and textbooks,
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a single teacher can change kids maths relationship
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or other subjects and forever.
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What's the role of the parents in this picture?
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Let's go to another difficult subject.
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Yeah, that is a difficult subject.
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One study found that
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the amount of maths anxiety parents had
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predicted their child's achievement in school,
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but only if they helped with homework.
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Oh, that's so funny.
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There were some interesting implications for this.
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I mean, you can see how it works.
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If you have maths anxiety
link |
and you're helping your kids with homework,
link |
you're probably communicating things like,
link |
I was terrible at this at school
link |
and that's how it gets passed on to kids.
link |
So one implication is if you have a really bad relationship,
link |
maths, you hate maths, you have maths anxiety,
link |
and maths anxiety just don't do maths homework with your kids.
link |
But we have a, on our website,
link |
we have a little sheet for parents
link |
of ways to interact around maths with your kids.
link |
That's ucubed.org.
link |
That's ucubed.org, yes.
link |
So one of the things I say to parents
link |
from when I do parent presentations is,
link |
even if you hate maths,
link |
you need to just fake it with your kids.
link |
You should be always endlessly optimistic
link |
and happy about doing maths.
link |
I'm always curious about this.
link |
So I, you know, I hope to have kids one day.
link |
I don't have kids currently.
link |
Are parents okay with like sucking at math
link |
and then trying to get their kid to be better
link |
than them, essentially?
link |
Like, is that difficult thing for a lot of parents?
link |
To have like, it's almost like an ego thing.
link |
Like, I never got good at this
link |
and I probably should have.
link |
And yeah, I mean, to me that you want to celebrate that,
link |
but I know a lot of people struggle with that.
link |
Like coaches in sports,
link |
to make an athlete become better than them,
link |
it can be hard on the ego.
link |
So is that, do you experience the same with parents too?
link |
I think, I mean, I haven't experienced parents
link |
worrying that their kids will be better than them.
link |
I have experienced,
link |
I have experienced parents
link |
just having a really bad relationship with maths
link |
and, you know, not wanting to help,
link |
not knowing how to help, saying things.
link |
Like another study showed that when mothers say
link |
to their daughters, I was bad at maths in school,
link |
their daughters achievement goes down.
link |
So we know that kids pick up on these messages
link |
and which is why I say you should fake it.
link |
But also I know that lots of people have just had
link |
a really bad relationship with maths,
link |
even successful people.
link |
The undergrads I teach at Stanford
link |
have pretty much always done well in maths,
link |
but they come to Stanford thinking maths
link |
is a set of methods to memorize.
link |
And so, so do many parents believe that.
link |
There's one method that you memorize
link |
and then you reproduce it.
link |
So until people have really had an experience
link |
of what I think of as the other maths,
link |
but until they've really seen
link |
that it's a really different subject,
link |
it's hard for them to be able to shift their kids
link |
to see it differently.
link |
Is there for a teacher, if we're to like systematize it,
link |
is there something teachers can do
link |
to do this more effectively?
link |
So you mentioned the textbook.
link |
So what are the additional things you can add
link |
on top of this whole old school traditional way
link |
of teaching that can improve the process?
link |
So I do think there's a way of teaching maths
link |
that changes everything for kids and teachers.
link |
So I'm one of five writers of a new framework
link |
for the State of California new maths framework.
link |
It's coming out next year.
link |
And we are recommending through this maths framework
link |
that people teach in this way.
link |
It's called teaching to big ideas.
link |
So at the moment, people have standards
link |
that have been written.
link |
And then textbooks have taken these standards
link |
and made not very good questions.
link |
And if you look at the standards,
link |
like I have some written down here,
link |
just reading the standards,
link |
it makes maths seem really boring and uninspiring.
link |
What are the kind of, can you give a few examples?
link |
So this is an interesting example.
link |
In third grade, there are three different standards
link |
about unit squares.
link |
So this is one of them.
link |
A square with side length one unit called a unit square
link |
is said to have one square unit of area
link |
and can be used to measure area.
link |
And that's something you're expected to learn.
link |
That is something, so that's a standard.
link |
The textbook authors say,
link |
oh, I'm gonna make a question about that.
link |
And they translate the standards into narrow questions.
link |
And then you measure success
link |
by your ability to deliver on these standards.
link |
So the standards themselves,
link |
I think of maths and many people think maths in this way
link |
as a subject of like a few big ideas
link |
and really important connections between them.
link |
So like, you could think of it as like a network map
link |
of ideas and connections.
link |
And what standards do is they take that beautiful map
link |
and they chop it up like this
link |
into lots of little pieces
link |
and they deliver the pieces to schools.
link |
And so teachers don't see the connections between ideas
link |
So anyway, this is a bit of a long way of saying
link |
that what we've done in this new initiative
link |
is we have set out maths as a set of big ideas
link |
and connections between them.
link |
So this is a grade three.
link |
So instead of there being 60 standards,
link |
we've said, well, you can pull these different standards
link |
to get in with each other
link |
and also value the ways these are connected.
link |
And by the way, for people who are just listening,
link |
we're looking at a small number of like big concepts
link |
within mathematics, square towels,
link |
measuring fraction, shape and time
link |
and then how they're interconnected.
link |
And so the goal is for, this is for grade three,
link |
Yeah, and so we've set out for the state of California,
link |
the whole of mathematics K10
link |
as a set of big ideas and connections.
link |
So we know that teachers, it works really well
link |
if they say, okay, so a big idea in my grade is measuring.
link |
And instead of reading five procedural statements
link |
that involve measuring,
link |
they think, okay, measuring is a big idea.
link |
What rich deep activity can I use
link |
that teaches measuring to kids?
link |
And as kids work on these deep rich activities,
link |
maybe over a few days,
link |
turns out a lot of maths comes into it.
link |
So we're recommending that let's not teach maths
link |
according to all these multiple, multiple statements
link |
and lots and lots of short questions.
link |
Instead, let's teach maths by thinking about
link |
what are the big ideas
link |
and what are really rich deep activities
link |
that teach those big ideas.
link |
So that's the like how you teach it
link |
and maximize learning.
link |
What about like from a school district perspective,
link |
like measuring how well you're doing,
link |
you know, grades and tests and stuff like that.
link |
Do you throw those out or is it possible?
link |
I am not a fan of grades and tests myself.
link |
I think grades are fine
link |
if they're used at the end of a course.
link |
So at the end of my maths course, I might get a grade
link |
because a grade is meant to be a summative measure.
link |
It kind of describes your summative achievement.
link |
But the problem we have in maths classrooms across the US
link |
is people use grades all the time, every week
link |
or every day even.
link |
My own kids, when they went through high school,
link |
technology has not helped with this.
link |
When they went through high school,
link |
they knew they were being graded
link |
for everything they did, everything.
link |
And not only were they being graded for everything,
link |
but they could see it in the grade book online
link |
and it would alter every class they went into.
link |
So this is the ultimate what I think of
link |
as a performance culture.
link |
You're there to perform, somebody's measuring you,
link |
you see your score.
link |
So I think that's not conducive for deep learning.
link |
And yes, have a grade at the end of the year,
link |
but during the year,
link |
you can assess kids in much better ways.
link |
Like teachers can, a great way of assessing kids
link |
is to give them a rubric that kind of outlines
link |
what they're learning over the course of a unit
link |
So kids can actually see the journey they're on.
link |
Like this is what we're doing mathematically.
link |
Sometimes they self assess on those units.
link |
And then teachers will show what the kids can do
link |
with a rubric and also write notes.
link |
Like in the next few weeks,
link |
you might like to learn to do this.
link |
So instead of kids just thinking about I'm an A kid
link |
or a B kid or I have this letter attached to me,
link |
they're actually seeing mathematically what's important.
link |
And they're involved in the process
link |
of knowing where they are mathematically.
link |
At the end of the year, sure, they can have a grade,
link |
but during the year,
link |
they get these much more informative measures.
link |
I do think this might be more for college,
link |
Some of the best classes I've had
link |
is when I got a special set aside,
link |
like the professor clearly saw that I was interested
link |
in some aspect of a thing.
link |
And then a few in mind and one in particular,
link |
he said that he kind of challenged me.
link |
So this is outside of grades and all that kind of stuff
link |
that basically it's like reverse psychology.
link |
I don't think this can be done.
link |
And so I gave everything to do that particular thing.
link |
So this happened to be in an artificial intelligence class.
link |
But I think that like special treatment
link |
of taking students who are especially like excellent
link |
at a particular little aspect
link |
that you see their eyes light up.
link |
I often think like maybe it's tempting for a teacher
link |
to think you've already succeeded there,
link |
but they're actually signaling to you
link |
that like you could really launch them on their way.
link |
that's too much to expect from teachers,
link |
I think to pay attention to all of that
link |
because it's really difficult.
link |
But I just kind of remember who are the biggest,
link |
the most important people in the history
link |
of my life of education.
link |
And it's those people that who really didn't just like
link |
inspire me with their awesomeness, which they did,
link |
but also just they pushed me a little,
link |
like they gave me a little push.
link |
And that requires focusing on the quote unquote,
link |
excellent students in the class.
link |
I think what's important though is teachers
link |
to have the perspective that they don't know
link |
who's gonna be excellent at something
link |
before they give out the activity.
link |
And in our camp classes that we ran,
link |
sometimes students would finish ahead of other students.
link |
And we would say to them,
link |
can you write a question that's like this, but different?
link |
Oh, and over time,
link |
we encourage them to like extend things further.
link |
I remember we were doing one activity
link |
where kids were working out the borders of a square
link |
and how big this border would be in different case sizes.
link |
And one of the boys came up at the end of the class
link |
and said, I've been thinking about
link |
how you do this with a Pentagon.
link |
And I said, that's fantastic.
link |
what does it look like with Pentagon?
link |
Go find out, see if you can discover.
link |
So I didn't know he was gonna come up and say that.
link |
And I didn't have in my head,
link |
like this is the kid who could have this extension task,
link |
but you can still do that as a teacher
link |
when kids get excited about something
link |
or they're doing well in something,
link |
have them extend it, go further.
link |
And then you also, like this is like teacher and coach,
link |
you could say it in different ways to different students.
link |
Like for me, the right thing to say is,
link |
almost to say, I don't think you could do this.
link |
Like that's what I need to hear.
link |
It's like, no, I, you know, there's immediate push,
link |
but with some people, if they're a little bit more,
link |
I mean, it's all has to do with upbringing,
link |
how your genetics is.
link |
They might be much more, that might break them.
link |
Yeah, that might break them.
link |
And so you have to be also sensitive to that.
link |
I mean, teaching is really difficult.
link |
It is really difficult.
link |
For this very reason.
link |
So what is the best way to teach math,
link |
to learn math at those early few days
link |
when you just want to capture them?
link |
Actually, there's a video of me doing this on our website
link |
that I love when I first meet students.
link |
And this is what I do.
link |
I show them a picture.
link |
This is the picture I show them.
link |
And it's a picture of seven dots like this.
link |
And I show it for just a few seconds and I say to them,
link |
I'd like you to tell me how many dots there are,
link |
but I don't need to count them.
link |
I want to group the dots.
link |
And I show it them and then I take it away
link |
before they've even had enough time to count them.
link |
And then I ask them, so how did you see it?
link |
And I go around the room and amazingly enough,
link |
there's probably 18 different ways of seeing these seven dots.
link |
And so I ask people, tell me how you grouped it.
link |
And some people see it as like an outside hole
link |
with a center dot.
link |
Some people see like stripes of lines.
link |
Some people see segments.
link |
And I collect them all and I put them on the board.
link |
And at the end I say, look at this,
link |
we are a class of 30 kids and we saw these seven dots
link |
in 18 different ways.
link |
There's actually a mathematical term for this.
link |
It's called groupitizing.
link |
It's kind of cool.
link |
So turns out though, that how well you groupitize predicts
link |
how well you do in maths.
link |
Is it a raw talent or is it just something
link |
that you can develop?
link |
I don't think it's wrong.
link |
I don't think you're born groupitizing, I think.
link |
But some kids have developed that ability, if you like.
link |
And you can learn it.
link |
So this to me is part of how wrong we have maths
link |
that we think to tell whether a kid's good at maths,
link |
we're gonna give them a speed test on multiples.
link |
But actually seeing how kids group dots
link |
could be a more important assessment
link |
of how well they're gonna do in maths.
link |
Anyway, I diverge.
link |
What I like to do though when I start off with kids
link |
I'm gonna give you maths problems.
link |
I'm gonna value the different ways you see them.
link |
And turns out you can do this kind of problem
link |
asking people how they group dots with young children
link |
or with graduate students.
link |
And it's engaging for all of them.
link |
Is a, you talk about creativity a little bit
link |
and flexibility in your book limitless.
link |
What's the role of that?
link |
So it sounds like there's a bit of that kind of thing
link |
involved in groupitizing.
link |
Yeah, I love this term.
link |
So what would you say is the role of creativity
link |
and flexibility in the learning of math?
link |
I think what we know now is that
link |
what we need for this 21st century world we live in
link |
is a flexible mind.
link |
Schools should not really be about teaching kids
link |
particular methods, but teaching them to
link |
approach problems with flexibility.
link |
Being creative, thinking creatively is really important.
link |
So people don't think the words maths and creativity
link |
come together, but that's what I love about maths
link |
is the creative different ways you can see it.
link |
And so helping our kids.
link |
There's a book I like a lot by, been by physicists.
link |
You probably know this book called Elastic.
link |
You might know it.
link |
And it's about how we want elastic minds,
link |
same kind of thing, flexible creative minds.
link |
And schools do very little on developing that kind of mind.
link |
They do a lot of developing the kind of mind
link |
that a computer now does for us.
link |
Memorization, doing procedures,
link |
a lot of things that we spend a lot of time in school on
link |
in the world when kids leave school,
link |
a computer will do that.
link |
And better than they will.
link |
But that creative, flexible thinking,
link |
we're kind of at ground zero at computers
link |
being able to engage in that thinking.
link |
Maybe we're a little above ground zero,
link |
but the human brain is perfectly suited
link |
for that creative, flexible thinking.
link |
That's what humans are so great at.
link |
So I would like to balance the shift in schools.
link |
Maybe you still need to do some procedural kind of thinking,
link |
but there should be a lot more of that
link |
creative, flexible thinking.
link |
And what's the role of other humans in this picture?
link |
So collaborative learning.
link |
So brainstorming together.
link |
So creativity as it emerges
link |
from the collective intelligence of multiple humans.
link |
Yeah, super important.
link |
And we know that also helps develop your brain,
link |
that social side of thinking.
link |
And I love mathematics collaboration
link |
where people build on each other's ideas
link |
and they come up with amazing things.
link |
I actually taught a hundred students calculus
link |
at Stanford recently, undergrads.
link |
And we taught them to collaborate.
link |
So these students came in Stanford
link |
and most of them were against collaboration in maths.
link |
This is before COVID in person?
link |
Yeah, it was just before COVID hit.
link |
Sorry, you said they're against.
link |
Yeah, so it's really interesting.
link |
So they'd only experienced maths individually
link |
in a kind of competitive individual way.
link |
And if they had experienced it as group work,
link |
it had been a bad experience.
link |
Like maybe they were the one who did it all
link |
and the others didn't do much.
link |
So they were kind of against collaboration.
link |
They didn't see any role for it in maths.
link |
And we taught them to collaborate.
link |
And it was hard work because as well as the fact
link |
that they were kind of against collaboration,
link |
they came in with a lot of like social comparison thinking.
link |
So I'm in this room with other Stanford undergrads
link |
and they're better than me.
link |
So when we sent them to work on a maths problem together,
link |
the first one was kind of a disaster
link |
because they put all like, they're better than me,
link |
they're faster than me.
link |
They came up with something I didn't come up with.
link |
So we taught them to let go of that thinking
link |
and to work well together.
link |
And one of the things we did, we decided,
link |
we wanted to do a pre and post test
link |
at the end of this teaching, it was only four weeks long.
link |
But we knew, we didn't want to give them
link |
like a time test of individual work.
link |
So we gave them an applied problem to do at the beginning.
link |
And we gave them to do pairs together.
link |
And we gave each of them a different colored pen
link |
and said, work on this activity together
link |
and keep using that pen.
link |
So then we had all these pieces of student work
link |
and what we saw was they just worked
link |
on separate parts of the paper.
link |
So there's a little like red pen section
link |
and a green pen section.
link |
And they didn't do that well on it.
link |
Even though it was a problem
link |
that middle or high school kids could do,
link |
but it was like a problem solving kind of problem.
link |
And then we gave them the same one to do at the end,
link |
gave them the same colors
link |
and it actually they had learned to collaborate.
link |
And not only were they collaborating the second time
link |
around, but that boosted their achievement.
link |
And the ones who collaborated did better on the problem.
link |
Collaboration is important having people
link |
and what was so eye opening for these undergrads
link |
and they talked about it in lovely ways
link |
was I learned to value other people's thinking on a problem.
link |
And I learned to value that other people saw it
link |
in different ways.
link |
And it was quite a big experience for them
link |
that they came out thinking,
link |
I can do maths with other people,
link |
people can see it differently,
link |
we can build on each other's ways of thinking.
link |
I got a chance to,
link |
I don't know if you know who Daniel Kahneman is,
link |
got a chance to interact with him.
link |
And like the first,
link |
cause he had a few but one famous collaboration
link |
throughout his life with Tversky.
link |
And just like, you know,
link |
he hasn't met me before in person,
link |
but just the number of questions he was asking
link |
and just the curiosity.
link |
So I think one of the skills,
link |
the collaboration itself is a skill.
link |
And I remember my experience with him was like,
link |
okay, I get why you're so good at collaboration
link |
because he was just extremely good at listening
link |
and genuine curiosity about how the other person
link |
thinks about the world, sees the world.
link |
And then together he pulled me in in that particular case.
link |
He doesn't know in particular like
link |
that much about autonomous vehicles,
link |
but he kept like asking all of these questions.
link |
And then like 10 minutes in,
link |
we're together trying to solve the problem
link |
of autonomous driving.
link |
And like, and that,
link |
I mean, that's really fulfilling,
link |
that's really enriching,
link |
but it also in that moment made me realize
link |
it's kind of a skill.
link |
Is you have to kind of put your ego aside,
link |
put your view of the world aside
link |
and try to learn how the other person sees it.
link |
And the other thing you have to put aside
link |
is this social comparison thinking.
link |
If you are sitting there thinking,
link |
wow, that was an amazing idea.
link |
He's so much better than I am.
link |
That's really gonna stop you taking on
link |
the value of that idea.
link |
And so there's a lot of that going on
link |
between these Stanford students when they came.
link |
And trying to help them let go of that.
link |
One of the things I've discovered
link |
just because being a little bit more in the public eye,
link |
how rewarding it is to celebrate others.
link |
And how much it's going to actually pay off
link |
So this kind of silo thinking of like,
link |
I want to prove to a small set of people around me
link |
that I'm really smart and do so
link |
by basically not celebrating how smart the other people are.
link |
That's actually maybe short term,
link |
it seems like a good strategy, but long term it's not.
link |
And I think if you practice at the student level
link |
and then at the career at every single stage,
link |
I think that's ultimately.
link |
I think that's a really good way of thinking about it.
link |
You mentioned textbooks and you didn't say it.
link |
Maybe textbooks isn't the perfect way to teach mathematics,
link |
but I love textbooks.
link |
They're like pretty pictures and they smell nice
link |
and they open, I mean, I talk about like physical.
link |
Some of them my greatest experiences
link |
have been just like,
link |
because they're really well done
link |
when we're talking about basic like high school,
link |
calculus, biology, chemistry.
link |
Those are like, those are incredible.
link |
It's like Wikipedia, but with color and nice little stuff.
link |
You must have seen some good textbooks
link |
if they had pretty pictures in color.
link |
Yeah, I mean, I remember,
link |
I guess it was very, very standard,
link |
like AP calculus, AP biology, AP chemistry.
link |
I felt those are like some of the happiest days of my life
link |
in terms of learning was high school
link |
because it was very easy, honestly.
link |
It felt hard at the time,
link |
but you're basically doing a whirlwind tour
link |
of all of science.
link |
Without having to pick.
link |
You do literature, you do like Shakespeare,
link |
your calculus, biology, physics, chemistry,
link |
what else, anatomy, physiology, computer science.
link |
Without like, nobody's telling you what to do.
link |
Your life, you're just doing all of those things.
link |
That's a good thing, you're right.
link |
But I remember the textbooks weren't,
link |
I mean, maybe I'm romanticizing the past,
link |
but I remember they weren't, they're pretty good.
link |
But so you think, what role do you think they play still?
link |
And like in this more modern digital age,
link |
what's the best materials
link |
with which to do these kinds of educations?
link |
Well, I'm intrigued that you had such a good experience
link |
I mean, I can remember loving some textbooks
link |
I had when I was learning and I love books.
link |
I love to pick up books and look through them.
link |
But a lot of maths textbooks
link |
are not good experiences for kids.
link |
They, we have a video on our website
link |
of the kids who came to our camp
link |
and one of the students says,
link |
in maths, you have to follow the textbook.
link |
The textbook's kind of like the Bible.
link |
You have to follow it.
link |
And every day it's slightly different.
link |
Like on Monday you do 2.3.2
link |
and on Tuesday you do 2.3.3 and on Wednesday.
link |
And you never go off that.
link |
That's like every single day.
link |
And that's not inspiring for a lot of the kids.
link |
So one of the things they loved about our camp
link |
was just that there were no books.
link |
Even though we gave them sheets of paper instead,
link |
they still felt more free
link |
because they weren't just like trotting through exercises,
link |
Like what a textbook allows you is like,
link |
you're, the very thing you said they might not like
link |
the 2.3, 2.3, it feels like you're making progress
link |
and like it's little celebrations
link |
because you do the problem and it seems really hard
link |
and you don't know how to do it.
link |
And then you try and try
link |
and then eventually succeed
link |
and then you make that little step and further progress
link |
and then you get to the end of a chapter
link |
and you get to like, it's closure.
link |
You're like, all right, I got that figured out
link |
and then you go on to the next chapter.
link |
I mean, I think it could be in a textbook.
link |
You can have a good experience with a textbook.
link |
But what's really important is what is in that textbook.
link |
What are you doing inside it?
link |
And I mean, I grew up in England
link |
and in England, we learn maths.
link |
We don't have this separation of algebra and geometry.
link |
And I don't think any other country
link |
apart from the US has that.
link |
But I look at kids in algebra classes
link |
where they're doing algebra for a year
link |
and I think I would have been pretty bored doing that.
link |
By the way, can we analyze your upbringing real quick?
link |
Why do British folks call mathematics maths?
link |
Why is it the plural?
link |
Is it because of everything you're saying
link |
or is it a bunch of sub disciplines?
link |
Yeah, I mean, mathematics is supposed to be the different maths
link |
Whether you think of that as topics like geometry
link |
and probability or I think of it as maths.
link |
It's just multidimensional, lots of ways.
link |
But that's why it was called mathematics
link |
and then it was shortened to maths.
link |
And then for some reason it was just math in the US.
link |
But to me, math has that more singular feel to it.
link |
And there's an expression here which is do the math,
link |
which basically means do a calculation.
link |
That's what people mean by do the math.
link |
So I don't like that expression
link |
because no math could be anything.
link |
It doesn't have to be a calculation.
link |
So yeah, I like maths
link |
because it has more of that broad feel to it.
link |
Yeah, I love that.
link |
Maths kind of emphasize the multidimensional,
link |
like the variety of different disciplines,
link |
different approaches, yeah.
link |
But outside of the textbook,
link |
what do you see broadly being used?
link |
You mentioned Sebastian Thrun and Moocs,
link |
Do you think that's an effective set?
link |
I mean, online, having great teachers online
link |
obviously extends those teachers to many more people
link |
and that's a wonderful thing.
link |
I have quite a few online courses myself.
link |
I got the bug working with Sebastian
link |
when he had released his first Mooc.
link |
And I thought, maybe I could do one in maths education
link |
and I didn't know if anybody would take it.
link |
I remember releasing it that first summer
link |
and it was a free online class
link |
and 30,000 maths teachers took it that first summer
link |
and they were all talking about it with each other
link |
and sharing it and it was like took off.
link |
In fact, it was that Mooc that got me to create Ucubed
link |
with Kathy Williams, who's the cofounder
link |
because people took the Mooc and then they said,
link |
Like I finished, what can I have next?
link |
And so that was where we made our website,
link |
but so yeah, I think online education can be great.
link |
I do think a lot of the Moocs don't have great pedagogy.
link |
They're just a talking head
link |
and you can actually engage people in more active ways
link |
even in online learning.
link |
So I learned from the Udacity principle
link |
when I was working at Udacity,
link |
never to talk more than like five minutes.
link |
And then to ask people to do something.
link |
So that's the sort of pedagogy of the online classes I have.
link |
There's a little bit of presenting something
link |
and then people do something and there's a little bit more
link |
because I think if you have a half hour video,
link |
you just switch off and start doing other things.
link |
So the way Udacity did it is like five, 10 minute
link |
like bit of teaching with some visual stuff perhaps
link |
and then there's like a quiz almost.
link |
Then you answer a question, yeah.
link |
Yeah, that's really effective.
link |
You mentioned Ucubed.
link |
So what's the mission?
link |
You mentioned how it started,
link |
but where are you at now and what's your dream with it
link |
or what are the kind of things
link |
that people should go and check out on there?
link |
Yeah, we started Ucubed,
link |
I guess it was about five years ago now
link |
and we've had over 52 million visitors to the site.
link |
So I'm very happy about that.
link |
And our goal is to share good ideas for teaching
link |
with teachers, students, parents in maths
link |
and to help, we have a sort of sub goal
link |
of a raising maths anxiety, that's important to us,
link |
but also to share maths as this beautiful creative subject.
link |
And it's been really great.
link |
We have lessons on the site,
link |
but one of the reasons I thought this was needed
link |
is there's a lot of knowledge in the academy
link |
about how to teach maths.
link |
Well, loads and loads of research and journals
link |
and lots of things written up,
link |
but teachers don't read it.
link |
They don't have access to it.
link |
They're often behind pay walls.
link |
They're written in really inaccessible ways,
link |
so people wouldn't want to read them or understand them.
link |
So this I see is a big problem.
link |
You have this whole industry of people
link |
finding out how to teach well,
link |
not sharing it with the people who are teaching.
link |
So that's where we made U cubed.
link |
And instead of just putting articles up,
link |
saying here's some things to read about how to teach well,
link |
we translated what was coming from research
link |
into things that teacher could use.
link |
So lessons, there were videos to show kids
link |
and there were tips for parents.
link |
There were all sorts of things on the site.
link |
And it's been amazing as we took inspiration
link |
from the week of code, which got teachers
link |
to focus on coding for a week.
link |
And we have this thing called the week of inspirational maths.
link |
And we say, just try it for a week.
link |
Just give us one week and try it and see what happens.
link |
And so it's been downloaded millions of times.
link |
Teachers use it every year.
link |
They start the school year with it.
link |
And what they tell us is it was amazing.
link |
The kids lights were on.
link |
They were excited.
link |
And then the week finished and I opened my textbooks
link |
and the lights went out and they were not interested.
link |
Yeah, but getting that first inspiration is still powerful.
link |
I wish, I mean, my, what I would love
link |
is if we could actually extend that for the whole year.
link |
We're a small team at Stanford
link |
and we're trying to keep up with great things
link |
to put on the site.
link |
We have the capacity to produce these creative visual maths
link |
tasks for every year group for every day.
link |
But I would love to do that.
link |
How difficult is it to do?
link |
I mean, it's to come up with visual formulations
link |
of these big important topics you need to think about
link |
in a way that you could teach.
link |
I mean, we can do it.
link |
We actually, we went from the week of inspirational maths
link |
and we made K8 maths books with exactly that.
link |
Big ideas, rich activities, visuals.
link |
We just finished the last one.
link |
We've been doing it for five years
link |
and it's been exhausting and we just finished.
link |
So now there's a whole K8 set of books
link |
and they're organized in that way.
link |
These are the big ideas here are rich deep activities.
link |
They're not though what you can do every day for a year.
link |
So some teachers use them as a kind of supplement
link |
to their boring textbook.
link |
And some people have said, okay, this is the year.
link |
This book tells us what the year is
link |
and then we'll supplement these big activities with.
link |
So they're being used and teachers really like them
link |
and are really happy about them.
link |
I just always want more.
link |
And I guess one of the things I would like for Ucubed,
link |
one of my personal goals is that every teacher of maths
link |
knows about Ucubed.
link |
lots of teachers who come to us are really happy
link |
they found it, but there's a lot of other teachers
link |
who don't know that it exists.
link |
I hope this helps.
link |
From a student perspective
link |
and not in the classroom, but at home studying.
link |
Is there some advice you can give
link |
on how to best study mathematics?
link |
So what's the role of the student outside the classroom?
link |
Yeah, I think one thing we know is
link |
a lot of people, when they review material,
link |
whether it's maths or anything else,
link |
don't do it in the best way.
link |
I think a problem a lot of people have is they read through
link |
maybe a teacher's explanation or a way of doing maths
link |
and it makes sense.
link |
And they think, oh yeah, I've got that and they move on.
link |
But then it's not until you come to try and work on something
link |
and do a problem that you actually realize
link |
you didn't really understand it,
link |
it just seemed to make sense.
link |
So I would say this is also something
link |
that neuroscience does talk about,
link |
to keep giving yourself questions
link |
is a really good way to study
link |
rather than looking through lots of material.
link |
It's always like giving yourself lots of tests
link |
is a good way to actually deeply understand things
link |
and know what you do and you don't understand.
link |
So would the questions be in the form of
link |
the material you're reviewing is the answer to that question
link |
or is it almost like beyond,
link |
it's the polygon thing they mentioned from a square,
link |
is it almost like I wonder what is the bigger picture
link |
I was kind of asking of like how is this extended and so on?
link |
Yeah, that would be great.
link |
And it's a similar, I mean, a question I get asked a lot
link |
is about homework, what is a good thing
link |
for kids to do for homework.
link |
And one of the recommendations I give is
link |
to not have kids just do lots of questions for homework,
link |
but to actually ask them to reflect on what they've learned,
link |
like what was the big idea you learned today?
link |
Or what did you find difficult?
link |
What did you struggle with?
link |
What was something that was exciting?
link |
Then kids go home and they have to kind of reflect
link |
A lot of times, I don't know if you have this experience
link |
as a math student, lots of people do.
link |
Kids are going through math's questions,
link |
they're successful, they get them right,
link |
but they don't even really know what they're about.
link |
And a lot of kids go through many years of math like that,
link |
doing lots of questions, but really knowing
link |
what even the topic is or what it's about,
link |
what it's important for.
link |
So having students go back and think at the end of a day,
link |
what was the big idea from this math lesson?
link |
Why is it important?
link |
Where would I find that in real life?
link |
Those are really good questions
link |
for kids to be thinking about.
link |
It's probably for everybody to be thinking about.
link |
I think most of us go through life
link |
never asking the bigger question,
link |
almost like those layers of why questions
link |
that kids ask when they're very young,
link |
we need to keep doing that.
link |
Like that's the, whatever the term is,
link |
you call first principles thinking,
link |
some people call it that,
link |
which is like, why are we doing it this way?
link |
So one nice thing is to do that
link |
because there's usually a good answer.
link |
Like the reason we did it this way
link |
is because it works for this reason.
link |
But then if you want to do something totally novel,
link |
is you'll say, well, we've been doing it this way
link |
because of historical reasons,
link |
but really this is not the best way to do it.
link |
There might be other ways.
link |
And that's how invention happens.
link |
And then you get, you know,
link |
that's really useful in every aspect of life,
link |
like choosing your career, choosing your,
link |
I don't know, where you live,
link |
who you're like romantic partner is like everything.
link |
And I think it probably starts doing that in math class.
link |
That would be good if we started doing that.
link |
I mean, I wonder, I probably didn't do very much of that
link |
for most of my education, asking why,
link |
except for later, much later in the subjects
link |
on like grad school when you're doing research on them.
link |
When your first task of doing something novel using this
link |
or solving a problem really outside the classroom,
link |
they have to publish on it.
link |
It's the first time you think, wait,
link |
why are these things interesting, useful,
link |
which are the things that are useful?
link |
And yeah, I guess that would be nice
link |
if we did that much earlier, that the quest of invention.
link |
I mean, one of the sad pieces of research data
link |
I think about is the questions kids ask
link |
in school goes down like in a linear, you know,
link |
progression from in the early years,
link |
you can't stop kids asking those questions,
link |
but they learn not to ask the questions.
link |
I think you told somewhere about an early memory
link |
you had in your own education where you asked the question,
link |
or maybe that was an example you gave,
link |
but it was shut down.
link |
Oh yeah, you've listened to something I said, yeah.
link |
I don't remember where it was, but it caught me.
link |
Yeah, I remember it really vividly.
link |
Well, can you tell the memory?
link |
Yeah, it's funny, I can remember.
link |
It must have really impacted me in that moment
link |
because you know, how there's lots of hours of school,
link |
we don't remember at all, but anyway,
link |
I can remember where I was sitting and everything.
link |
I was in high school maths class,
link |
although they don't call it that in England,
link |
and the teacher said,
link |
and it was like the first class of this teacher's class,
link |
and he said, ask if you have any questions.
link |
So at one point I put my hand up,
link |
and I said, I have a question,
link |
and he said something like, that's your question.
link |
And I was like, okay, I'm not asking any more questions
link |
And it hit hard in a way where you didn't wanna,
link |
the lesson you learn from that, I'm not gonna ask.
link |
Yeah, that was absolutely the lesson I asked.
link |
That's the last question I'm asking.
link |
And that was, yeah, he was the chair of the maths department,
link |
I remember that really well.
link |
So maybe because of that experience,
link |
one of the things we encourage when we teach kids
link |
is asking questions.
link |
And we value it when they ask questions,
link |
and we put them up on walls and celebrate.
link |
It's funny, because I wish there was a feedback signal,
link |
because he probably, to put a positive spin on it,
link |
he probably didn't realize the negative impact
link |
he's had in that moment, right?
link |
See, this is probably when you're more mature
link |
I had an amazing professor named Alisha Kafande
link |
in computer science,
link |
and he would encourage questions,
link |
but then he would tell everybody
link |
how dumb their questions are.
link |
But it was done, I guess, if you say it with love
link |
and respect behind it,
link |
then it's more like a friendly, humorous encouragement
link |
for more questions.
link |
It's an art, right?
link |
Yeah, that's what teaching is very hard.
link |
You have to time it right, because that kind of humor
link |
is probably better for when you're in grad school,
link |
versus when you're in the early education.
link |
Well, and I guess kids or young people get
link |
whether somebody's doing it to be funny,
link |
or has it, I mean, this is why teachers so hard.
link |
Even your tone can be impactful.
link |
It's so sad, because for that particular human,
link |
the teacher, you could have just had a bad day,
link |
and one statement can have a profound negative impact.
link |
I know, sadly, that math,
link |
there's a lot of math teachers
link |
who have that kind of approach,
link |
and they, I think they're suffering from the fact
link |
that they think people are math people, not math people,
link |
and that comes across in their teaching.
link |
But on the flip side, one positive statement.
link |
That's right, that is the flip side of that.
link |
And I myself had one teacher who was really amazing
link |
for me in maths, and she kept me in the subject.
link |
I probably wouldn't have left it.
link |
She was, her name was Mrs. Marshall.
link |
She was my A level maths teacher.
link |
So I was in England, you do lots of subjects to your 16,
link |
and then you choose like three or four subjects.
link |
So I had chosen maths, and you go to higher levels.
link |
Probably equivalent more to a master's degree in the US,
link |
because you're more specialised.
link |
But anyways, she was my teacher,
link |
and for the first time in my whole career in maths,
link |
she would give us problems and tell us
link |
to talk about them with each other.
link |
And so here I was sitting there at like 17,
link |
talking with friends about how to solve a math problem.
link |
That was the change that she made,
link |
but it was profound for me,
link |
because like those calculus students,
link |
I started to hear other people's ways of thinking
link |
and seeing it, we would talk together
link |
and come up with solutions.
link |
And I was like, that was it, that changed maths for me.
link |
So it wasn't some kind of personal interaction with her.
link |
It was more like she was a catalyst
link |
for that collaborative experience.
link |
I mean, yeah, the many ways teachers can inspire kids.
link |
I mean, sometimes it's a personal message,
link |
but it can be your teaching approach
link |
that changes maths for kids.
link |
You know, Cal Newport, he wrote a book called Deep Work,
link |
and he's a mathematician,
link |
he's a theoretical computer scientist,
link |
and he talks about the kind of the focus required
link |
to do that kind of work.
link |
Is there something you can comment on?
link |
You know, we live in a world full of distractions.
link |
That seems like one of the elements that makes studying,
link |
and especially the studying of subjects
link |
that require thinking like maths does, difficult.
link |
Is there something from a student perspective,
link |
from a teacher perspective that encourages deep work
link |
that you can comment on?
link |
Yeah, I think giving kids really inspiring deep problems,
link |
and we have some on our website,
link |
is a really important experience for them.
link |
Even if they only do it occasionally,
link |
but it's really important.
link |
They actually realize,
link |
I give a problem out often when I'm working with teachers,
link |
and I say to them,
link |
all right, I'm gonna check in with you after an hour.
link |
And they were like, an hour?
link |
They think it's shocking.
link |
And then they work on this problem,
link |
and after an hour, I say, okay, how are we doing?
link |
They're like, an hour's gone by, how is this possible?
link |
And so everybody needs those rich, deep problems.
link |
Most kids go through their whole maths experience
link |
of however many years, never once working on a problem
link |
in that kind of deep way.
link |
So the undergrad class I teach at Stanford, we do that.
link |
We work on these deep problems every session.
link |
And the students come away going,
link |
okay, I never want to go back to that maths relationship
link |
I had where it was just all about quick answers.
link |
I just don't want to go back to that.
link |
And so we can all, all teachers can incorporate
link |
those problems in their classrooms.
link |
Maybe they don't do them every day,
link |
but they at least give kids some experience
link |
of being able to work slowly and deeply
link |
and to go to deeper places
link |
and not be told they've got five minutes
link |
to finish 20 questions.
link |
But part of it is also just the exercise
link |
of sitting there maintaining focus
link |
of prolonged periods of time.
link |
That's not often, I mean, that's a skill.
link |
It's a skill that also could be discouraging.
link |
Like if you don't practice it,
link |
just sitting down for 10 minutes straight
link |
and maintaining deep focus could be exceptionally challenging.
link |
Like if you're really thinking about a problem.
link |
And I think it's really important to realize
link |
that that's a skill that you can, just like a muscle,
link |
you can build, you can start with five minutes
link |
and goes to 10 minutes to 30 and to an hour.
link |
And to be successful, I think in certain subjects
link |
like mathematics, you want to be able to develop that skill.
link |
Otherwise, you're not going to get to the really rewarding
link |
experience of solving these problems.
link |
There was a survey done of kids in school
link |
where they were asked,
link |
how long will you work on a maths problem
link |
before you give up and decide it's not possible to solve it?
link |
And the result, on average, across the kids was two minutes.
link |
That's a bad sign, but that was a powerful sign
link |
that they need to learn to not give up so quickly.
link |
We mentioned offline,
link |
because we've been talking so much about visualization.
link |
Grant Sanderson, three little one brown.
link |
So he's inspired millions of people
link |
with exactly the kind of way of thinking
link |
that you've been talking about.
link |
Yeah, I love his work.
link |
Converting sort of mathematical concepts into visual,
link |
visually representing them,
link |
exploring them in ways that help you illuminate the concepts.
link |
What do you think is the role of that?
link |
So he uses mostly programmatic visualizations.
link |
So it's the thing I mentioned where there's like animations
link |
created by writing computer programs.
link |
Like what do you think?
link |
How scalable is that approach?
link |
But in general, what do you think about his approach?
link |
Yeah, I think it's amazing.
link |
I should work with him.
link |
I can share some of our visuals
link |
and he can make them in that amazing way.
link |
So part of his storytelling,
link |
part of his like is creating the visuals
link |
and then weaving a story with those visuals
link |
that kind of builds.
link |
There's also drama in it.
link |
You start with a small example
link |
and then all of a sudden there's a surprise.
link |
And it really, it makes you fall in love with the concept.
link |
He does talk about that.
link |
His sense is like some of the stuff,
link |
he doesn't feel like he's teaching.
link |
Like the core curriculum,
link |
which is something, you know,
link |
he sees himself as an inspirational figure.
link |
But because I think it's too difficult
link |
to kind of convert all of the curriculum
link |
into those elements.
link |
Right. And probably you don't need to.
link |
I mean, you, if people get to experience
link |
mathematical ideas in the way that he shares them,
link |
that will change them
link |
and it will change the way they think.
link |
And maybe they could go on
link |
to take some other mathematical idea
link |
and make it that beautiful.
link |
Well, he does that.
link |
There's a, he created a library called Manum
link |
and he open sourced it.
link |
And that library is the,
link |
people should check it out.
link |
It's written in Python
link |
and uses some of those same elements.
link |
Like it allows you to animate equations
link |
and animate little shapes.
link |
Like people that, you know,
link |
he has a very distinct style in his videos
link |
and what that resulted in,
link |
even though from a software engineer perspective,
link |
the code he released is not like
link |
super well documented or perfect,
link |
but him releasing that,
link |
now there's all of these people
link |
And the, to me personally,
link |
the coolest thing is to see like
link |
people they're not, you know,
link |
don't have like a million subscribers or something.
link |
They have just a few views in the video,
link |
but it just seems like the process of them
link |
creating a video where they teach
link |
is like transformative to them
link |
from a student perspective.
link |
It's the old Feynman thing,
link |
the best way to learn is to teach.
link |
And then him releasing that into the wild is,
link |
it shows that, that impact.
link |
I think just giving people that idea
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that you can do that with maths
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and other subjects,
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there's bound to be people all around
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who can create more, which is cool.
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So I recommend people do like JavaScript or Python.
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You can, you can build like visualizations
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of most concepts in high school math.
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You can do a lot of kinds of visualizations
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and doing that yourself.
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Plus, if you do that yourself,
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people will really love it.
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people love visualizations of math.
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Because they, I mean, it's something in us
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that loves patterns,
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loves figuring out difficult things
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and the patterns in there
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that are unexpected in some way.
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Have you ever noticed that hotels
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are always filled with patterns?
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I was just noticing at the hotel.
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I mean, now all of their carpets are pattern carpets
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and then they have patterns on the walls.
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We humans love the symmetry in patterns,
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the breaking of symmetry in patterns.
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And it's funny that we don't see mathematics
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as somehow intricately connected to that,
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I mean, that's one of the perspectives I love students to take,
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is to be a pattern seeker.
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Certainly in all of maths.
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I mean, you can think of all of maths
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as a kind of subject of patterns
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and not just visual patterns,
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but when you think about multiplying by five
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and the fact you can,
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if you're multiplying 18 times five,
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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 halve a number and double a 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
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and find patterns, it's really exciting.
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What do you think about, like, MIT OpenCourseWare
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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 the MOOC I did,
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was using that platform.
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See, you ultimately think, like, the Udacity models
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is a little bit more effective
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than just a plane to our lecture.
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I think there's definitely you can bring in good pedagogy
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into online learning.
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And I think the idea of putting things online
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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,
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sort of democratizing education and making it more equal,
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came about because they found that the people taking MOOCs
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tended to be the more privileged people.
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So that was, I think, there's still something to be found
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in that there's still more to be done
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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,
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that's a little free class that gives them...
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It's called How to Learn Maths.
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How to Learn Maths?
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It shows maths as this visual creative subject
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and it shares mindset and some brain science.
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And kids who take it do better in maths class.
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We've studied it with, like, randomized controlled trials
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and given it to middle school kids
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and other middle school kids who don't take it
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but are taught by the same teachers,
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so their teachers are the same.
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And the kids who take the online class
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end up 68% more engaged in their maths class
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and do better at the end of the year.
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So that's a little six session, 15 minute class
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and it changes kids maths relationship.
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So it is true that we can do that with some words
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that aren't, you know, it's not a huge change
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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,
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but in terms of their journey through education,
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through their career choices, through life,
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maybe middle school, high school, undergrad students,
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how to live a life that they can be proud of.
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I think if I were to give advice to people,
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especially young people,
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my advice would be to always, it sounds really corny,
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but always believe in yourself and know that you can achieve
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because although that sounds like obvious,
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of course we want kids to know that they can achieve things.
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I know that millions of kids who are in the school system
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have been given the message they cannot do things.
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And adults too, they have the idea,
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oh, I did okay in this, I went into this job
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because those other things I could never have done okay in.
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So actually when they hear,
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hey, maybe you could do those other things.
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Even adults think, maybe I can.
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And they go back and they encounter this knowledge
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and they relearn things and they change careers
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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.
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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,
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areas of their brain that aren't functioning well
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that have what we call special needs.
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Some of those people also go on to develop
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and do amazing things.
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So I think that really experiencing that,
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knowing that feeling, not just saying it,
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but knowing it deeply.
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You can learn anything is something I wish all people would have.
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It actually also applies when you've achieved some level of success too.
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What I find in my life with people that love me,
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when you achieve success, they keep celebrating your success
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and they want you to keep doing the thing that you were successful at
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as opposed to believing in that you can do something else.
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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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quite recently, which is sad to say,
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is how important it is to seek out,
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when you're younger to seek out mentors,
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to seek out the people,
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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 like grad school,
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you think you kind of land on a mentor,
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maybe you pick a mentor based on the topic they're interested in.
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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...
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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 love you deeply, but they set...
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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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And in that, if you're interested in mathematics,
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your parents are not that interested in it,
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don't listen to your parents on that one dimension.
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And if people tell you you can't do things,
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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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It's so sad the number of people who've had those negative messages from parents.
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In my limitless mind book,
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I interviewed quite a few people who'd been told they couldn't do math,
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sometimes by parents, sometimes by teachers.
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And fortunately, they had got other ideas at some point in their life
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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.
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However hard that might be to find those people.
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What do you hope the education system,
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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 I do actually see change happening.
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We were talking earlier that data science is now,
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of course, you can take in high school instead of Algebra II.
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And that's pretty amazing because that content was set out in 1892
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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.
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But when you look ahead and think about all that we know
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and all that we can offer kids in terms of technology,
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you've got to think that 100 years from now,
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education will be totally different to the way it is now.
link |
Maybe we won't have subject boundaries anymore
link |
because those don't really make much sense.
link |
It's interesting to think how certain tools like programming,
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maybe they'll be deeply integrated in everything we do.
link |
You would think that all kids are growing up learning to program and create.
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So I just think the system of schooling we have now,
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people call it a factory model,
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it's not designed to inspire creativity.
link |
And I feel like that will also change.
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People might look back on these days and think they were hilarious,
link |
but maybe in the future kids will be doing their own programming
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and they'll be able to learn things and find out things
link |
and create things even as they're learning.
link |
And maybe the individual subject boundaries will go.
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Data science itself coming into the education system
link |
kind of illustrates that because people realize
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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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And who teaches it?
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So it's already raising those kind of questions
link |
and 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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It's happening across the United States as we speak.
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I wonder how they got initiated.
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Like how does change happen in the education system?
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Is it just a few revolutionary leaders?
link |
It's been an interesting journey seeing data science take off actually.
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There was a course that was developed in 2014
link |
by some people who thought data science
link |
was a good idea for high schoolers.
link |
And then after some kids took the course
link |
and nothing bad happened to them,
link |
they went to college and people started to accept it more.
link |
And then this was a big piece of the change in California.
link |
The UC system communicated.
link |
They sent out an email last year to 50,000 high schools
link |
saying we now accept data science.
link |
Kids can take it instead of Algebra II.
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That's a perfectly legitimate college pathway.
link |
So that was like a big green light for a lot of schools
link |
who were like wondering about whether they could teach it.
link |
So I think it happens in small spaces and expands.
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And it's a lot of age.
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Then it goes viral.
link |
California is ahead, I think, in creating courses
link |
and having kids go through it.
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But suddenly when I last looked,
link |
there were 12 states that were allowing data science
link |
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 a change is happening.
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Joe, as I said, I think mathematics is truly a beautiful subject.
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And you having an impact on millions of people's lives
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by educating them, by inspiring teachers to educate
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in the ways that you've talked about,
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in multidimensional ways, in visual ways,
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I think is incredible.
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You're spreading beauty into the world.
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So I really, really appreciate that you
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have spent your valuable time with me today.
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Thank you for talking.
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It was really good to talk to you.
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Thanks for listening to this conversation with Joe Bowler.
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To support this podcast,
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please check out our sponsors in the description.
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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.