back to indexJo Boaler: How to Learn Math | Lex Fridman Podcast #226
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The following is a conversation with Jo Bowler,
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a mathematics educator at Stanford
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and co founder 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 Jo 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, more interesting,
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some of their visual, some of their algebraic,
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geometry, all that kind of stuff.
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Yeah, I mean, I always say
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that you can take any maths area and make it visual.
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And we say to teachers,
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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 it 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 complimentary?
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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 maths, 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, so we'll give you visuals
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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 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 we call it
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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
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at building that part of their brain
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of using visualization for intuition building,
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because you ultimately want to understand
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the like the deepest secret underneath this problem.
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And for that, you have to intuit your way there.
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And you mentioned offline that one of the ways
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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 is, you know,
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you have Einstein thinking about a train,
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you know, and the speed of light
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and you know, that kind of intuition is useful.
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You start to like imagine a physical world,
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like how does this idea manifest itself
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in the physical world?
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And then start playing in your mind
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with that physical world and think,
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is this going to be true?
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Is this going to be true?
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Einstein is well known for thinking visually.
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And people talk about how he really didn't want
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to go anywhere with problems
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without thinking about them visually.
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But the other thing you mentioned
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that sparked something for me 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
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of intuition 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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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
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of like solving the puzzle, like cracking it open
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to get the simple explanation of why things are wrong.
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Why things are the way they are,
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as opposed to, like you said, 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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It's hard, 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 is a great sort of mathematician,
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you might think of him, 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, it 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,
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for N equals two, N equals three,
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you start to play with it.
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And then in the modern day, which I loved doing is,
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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 way too much work
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because you have to click and move and stuff.
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I love to interact through writing programs,
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but 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,
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like color, all of those kinds of things
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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 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
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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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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 valuable to learning
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when the thing that you're doing is challenging,
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So a lot of people say 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 maths
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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 maths brain or you're not.
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So when they start to struggle,
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they think, oh, I don't have that maths brain.
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And then they will literally sort of switch off
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in their brain and things will go downhill from that point.
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So struggle becomes a lot easier
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and you're able to struggle if you don't have that idea,
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but you know that you 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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Can I ask a difficult question here?
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So there's kind of, I don't know what the right term is,
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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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So, 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
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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 and fell in love
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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, good that you fell in love about the force.
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Well, but 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 like, it's not improving.
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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 offhand
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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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It's silly. I didn't really think in those concrete terms,
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but I just thought I could be great at feeling.
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And then when you struggle,
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the belief that you could be great is like,
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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 I'd 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,
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you're talking about a lot of geometry, a lot more geometry.
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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
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over and over with geometry.
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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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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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that in the Soviet Union, that idea is really different.
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Now, the downside of that idea that anybody can get there
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if you work hard is that thought
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that if you're not getting there, it's your fault.
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And I would add something into that.
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I would say that anybody can get there,
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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 going to be able to keep the kids excited
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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
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in the early grades of elementary school.
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But fifth grade teachers are often like,
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okay, we're going to prepare you now for middle school
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and we're going to give you grades and lots of tests.
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And that's when kids start to feel really badly
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And so middle school years,
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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 going to 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,
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it said an extra sentence at the bottom
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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
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you put on the bottom of all kids work.
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I'm giving this feedback because 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 thinking,
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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
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to be saying to kids, 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, 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,
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cause I'm almost 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 that this person
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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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Yeah, and you're right.
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Silicon Valley where I live is filled with people
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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
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to do amazing work in creative ways.
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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 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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became quickly an outcast of the system.
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Like the education, especially early education system,
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Is there ways for an education system
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to support people like that?
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Is it this kind of multidimensional learning
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that you mentioned?
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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
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is a mathematician called Mary Mizikani.
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I don't know if you've heard of her.
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She won the Fields Medal.
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She was from Iran.
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First woman in the world
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to win the Fields Medal in mathematics.
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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
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thought she was an artist
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because she was always visualizing.
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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
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because this young woman spent like two hours
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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,
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wow, I could have brought her like 13 year old
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into this PhD defense.
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They would not recognize this as maths.
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But when Maryam Izzakhani 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,
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she also shared that when she was in school,
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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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She grew up in Iran.
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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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that's why people talk about like the one teacher
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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's like, 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,
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you have the power to truly change lives.
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And like 20 years from now,
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I feel as a medalist will walk up to you
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and say thank you.
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Yeah, 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 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
link |
the amount of maths anxiety parents had
link |
predicted their child's achievement in school,
link |
but only if they helped with homework.
link |
Oh, that's so funny.
link |
There are some interesting implications for this.
link |
I mean, you can see how it works.
link |
If you have maths anxiety
link |
and you're helping your kids with homework,
link |
you're probably communicating things like,
link |
oh, I was terrible at this at school.
link |
And that's how it gets passed on to kids.
link |
So one implication is
link |
if you have a really bad relationship with maths,
link |
you hate maths, you have maths anxiety,
link |
just don't do maths homework with your kids.
link |
But we have on our website,
link |
we have a little sheet for parents
link |
of ways to interact around maths with your kids and...
link |
That's ucubed.org.
link |
That's ucubed.org, yes.
link |
So one of the things I say to parents
link |
when I give 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 hope to have kids one day.
link |
I don't have kids currently.
link |
Are parents okay with like sucking at maths
link |
and then trying to get their kid
link |
to be better 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, you wanna 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 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 not wanting to help,
link |
not knowing how to help, saying things.
link |
Like another study showed that when mothers
link |
say to their daughters,
link |
I was bad at maths in school,
link |
their daughter's 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
link |
have just had 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 |
well, 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,
link |
if we were 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
link |
you can add on top of this whole old school
link |
traditional way 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, a 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 by your ability
link |
to deliver on these standards.
link |
So the standards themselves, I think of maths
link |
and many people think of maths in this way
link |
as a subject of like a few big ideas
link |
and really important connections between them.
link |
So 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 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 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, this is for grade three, for example.
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, they think,
link |
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 statements
link |
and lots and lots of short questions.
link |
Instead, let's teach maths
link |
by thinking about 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 |
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,
link |
I might get a grade because a grade is meant
link |
to be a summative measure.
link |
It kind of describes your summative achievement.
link |
But the problem we have in maths classrooms
link |
across the US is people use grades all the time,
link |
every week 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,
link |
what I think of 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,
link |
I'm an A kid or a B kid,
link |
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 like set aside,
link |
like the professor clearly saw that I was interested
link |
in some aspect of a thing.
link |
And then I have a few in mind and one in particular,
link |
when 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 |
And I don't know, 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
link |
like 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 |
Yeah, 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, can you write a question
link |
that's like this but different?
link |
Oh, and over time we encourage them
link |
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 how you do this
link |
And I said, that's fantastic.
link |
How do you, 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 like this is the kid
link |
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 almost to say,
link |
I don't think you could do this, this is too hard.
link |
Like that's what I need to hear.
link |
It's just like, no, 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 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 wanna capture them?
link |
I do something, actually there's a video of me
link |
doing this on our website that I love
link |
when I first meet students.
link |
And this is what I do.
link |
I show them a picture, 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 want you to count them.
link |
I want you 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
link |
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
link |
and we saw these seven dots 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
link |
predicts 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 raw.
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 |
is show them I'm gonna give you maths problems.
link |
I'm gonna value the different ways you see them.
link |
And it 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 |
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 |
So what would you say is the role of creativity
link |
and flexibility in the learning of maths?
link |
I think what we know now is that what we need
link |
for this 21st century world we live in is a flexible mind.
link |
School should not really be about teaching kids
link |
particular methods but teaching them
link |
to 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, there's a book I like a lot
link |
by physicists, 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, a lot of things
link |
that we spend a lot of time in school on.
link |
In the world, when kids leave school,
link |
a computer will do that 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 the balance to 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 creative,
link |
flexible thinking.
link |
And what's the role of other humans in this picture?
link |
So collaborative learning, so brainstorming together.
link |
So creativity as it emerges from the collective intelligence
link |
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 math.
link |
This is before COVID in person?
link |
Yeah, it was just before COVID hit.
link |
So 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 and it was hard work
link |
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 or...
link |
So when we set 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.
link |
It was only four weeks long, but we knew
link |
we didn't want to give them like a time test
link |
of individual work.
link |
So we gave them an applied problem to do at the beginning
link |
and we gave them to do in 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 that middle
link |
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 actually they had learned to collaborate.
link |
And not only were they collaborating the second time round,
link |
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
link |
saw it 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,
link |
but one famous collaboration throughout his life
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 |
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's,
link |
he pulled me in in that particular case.
link |
He doesn't know in particular,
link |
like 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, 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 |
Cause 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 |
Like 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 is going to actually pay off in the long term.
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
link |
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 level, 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
link |
and they smell nice and they open.
link |
I mean, I talk about like physical.
link |
Some of my greatest experiences have been just like,
link |
cause they're really well done.
link |
When we're talking about basic, like high school,
link |
calculus, biology, chemistry, those are like,
link |
those are incredible.
link |
It's like Wikipedia, but with color and a nice little.
link |
You must've seen some good textbooks
link |
if they had pretty pictures and color.
link |
Yeah, I mean, I remember,
link |
I guess it was very, very standard, like AP calculus,
link |
AP biology, AP chemistry.
link |
I felt those were like some of the happiest days of my life
link |
in terms of learning was high school.
link |
Cause it was very easy, honestly.
link |
It felt hard at the time,
link |
but you're basically doing a whirlwind tour
link |
of all the science.
link |
Without having to pick, you do literature,
link |
you do like Shakespeare, calculus, biology, physics,
link |
chemistry, what else?
link |
Anatomy, physiology, computer science,
link |
without like, nobody's telling you what to do with your life.
link |
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 with which to do
link |
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 I had
link |
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 is 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 you're,
link |
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 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 |
where it's a bunch of subdisciplines?
link |
Yeah, I mean, mathematics is supposed to be
link |
the different maths that you look at,
link |
whether you think of that as topics
link |
like geometry and probability,
link |
or I think of it as maths is just multi dimensional
link |
lots of ways, 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 |
doesn't have to be calculation.
link |
And 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 multi dimensional,
link |
like a variety of different subdisciplines,
link |
different approaches.
link |
Okay, but outside of the textbook,
link |
what do you see like 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 YouCubed
link |
with Kathy Williams, who's the co founder,
link |
because people took the MOOC and then they said,
link |
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 |
is 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, no, that's really effective.
link |
You mentioned Ucubed, so what's the mission?
link |
You mentioned how it started, but what's yeah,
link |
where are you at now?
link |
And what's your dream with it?
link |
What are the kind of things that people should go
link |
Yeah, we started Ucubed, I guess it was about five years
link |
ago now 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 and to help.
link |
We have a sort of sub goal of a raising maths anxiety.
link |
That's important to us, but also to share maths
link |
as this beautiful creative subject.
link |
And it's been really great.
link |
We have lessons on the site, but one of the reasons
link |
I thought this was needed is there's a lot of knowledge
link |
in the academy about how to teach maths well.
link |
Loads and loads of research and journals
link |
and lots of things written up, but teachers don't read it.
link |
They don't have access to it.
link |
They're often behind pay walls, they've written
link |
in really inaccessible ways.
link |
So people wouldn't want to read them or understand them.
link |
So this actually is a big problem.
link |
You have this whole industry of people finding out
link |
how to teach well, not sharing it with the people
link |
So that's why we made Ucubed.
link |
And instead of just putting articles up saying,
link |
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.
link |
As we took inspiration from the week of code,
link |
which got teachers to focus on coding for a week.
link |
And we have this thing called the week
link |
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, they were excited, they loved it.
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 |
It is, I wish, I mean, 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 haven't the capacity to produce
link |
these creative visual maths tasks
link |
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.
link |
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 YouCubed,
link |
one of my personal goals is that every teacher of maths
link |
knows about YouCubed.
link |
At the moment, lots of teachers who come to us
link |
are really happy they found it
link |
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 and not in the classroom
link |
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 a lot of people
link |
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
link |
is they read through maybe a teacher's explanation
link |
or a way of doing maths and it makes sense.
link |
And they think, oh yeah, I've got that.
link |
But then it's not until you come to try
link |
and work on something and do a problem
link |
that you actually realize you didn't really understand it,
link |
just seemed to make sense.
link |
this is also something that neuroscientists 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
link |
is the answer to that question?
link |
Or is it almost like beyond,
link |
it's the polygon thing they mentioned for a square.
link |
Is it almost like, I wonder what is the bigger picture?
link |
I was kind of asking like, how is this extended and so on?
link |
Yeah, that would be great.
link |
And it's a similar,
link |
I mean, a question I get asked a lot is about homework.
link |
What is a good thing for kids to do for homework?
link |
And one of the recommendations I give
link |
is 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 had this experience
link |
as a math student, lots of people do.
link |
Kids are going through math 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 maths like that,
link |
doing lots of questions,
link |
but that really knowing what even the topic is
link |
or what it's about, 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 maths 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 |
always those layers of why questions
link |
that kids ask when they're very young.
link |
We need to keep doing that.
link |
Like 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 |
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 |
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, that's really useful
link |
in every aspect of life, like choosing your career,
link |
choosing your, I don't know, where you live,
link |
who your 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 grad school when you're doing research on them.
link |
When you're first tasked with doing something novel
link |
using this or solving a problem
link |
really outside the classroom,
link |
they have to publish on it.
link |
It's the first time you think,
link |
wait, 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, the quest of invention.
link |
Yeah, yeah, I mean, one of the sad pieces of research data
link |
I think about is the questions kids ask in school goes down
link |
like in a linear 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 |
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 |
Or can you tell the memory?
link |
Yeah, I was, it's funny, I can remember.
link |
It must've really impacted me in that moment
link |
because you know how there's lots of hours of school
link |
you don't remember at all, but anyway,
link |
I can remember where I was sitting and everything.
link |
I was in a 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 and I said,
link |
I have a question, and he said something like,
link |
that's your question?
link |
And I was like, oh, okay,
link |
I'm not asking any more questions in this class.
link |
And that hit hard in a way where you didn't wanna,
link |
the lesson you learn from that is I'm not gonna ask.
link |
Yeah, that was absolutely 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, 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 |
If he only knew, see, this is probably
link |
when you're more mature in grad school.
link |
I had an amazing professor named Ali Shakafande
link |
in computer science, and he would get,
link |
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 show,
link |
if you say it with love and respect behind it,
link |
then it's more like a friendly, humorous encouragement
link |
for more questions.
link |
Yeah, it's an art, right, to do it, to write.
link |
And then you have to time it right
link |
because that kind of humor is probably better
link |
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 or, you know,
link |
I mean, this is why teaching is so hard.
link |
Even your tone can be impactful.
link |
It's so sad because for that particular human,
link |
the teacher, you just had a bad day,
link |
and one statement can have a profound negative impact.
link |
I know, sadly, that there's a lot of maths teachers
link |
who have that kind of approach,
link |
and 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 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 |
And she was my A level maths teacher.
link |
So I was in England.
link |
You do lots of subjects till you're 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 specialized.
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
link |
and tell us 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, and we would talk together
link |
and come up with solutions.
link |
And I was like, that was it.
link |
That changed maths for me.
link |
It wasn't some kind of personal interaction with her.
link |
It was more like she was the 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, 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 realise, I do, I give a problem out often
link |
when I'm working with teachers, and I say to them,
link |
all right, I'm gonna check in with you after an hour.
link |
And they're 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?
link |
How is this possible?
link |
And so everybody needs those like 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, okay,
link |
I never wanna go back to that maths relationship I had
link |
where it was just all about quick answers.
link |
I just don't wanna 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 and not be told
link |
they've got five minutes to finish 20 questions.
link |
Well, part of it is also just the exercise
link |
of sitting there and maintaining focus
link |
for 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 wanna be able to develop that skill.
link |
Otherwise you're not going to get
link |
to the really rewarding experience
link |
of solving these problems.
link |
There was a survey done of kids in school
link |
where they were asked, how long will you work
link |
on a maths problem before you give up
link |
and decide it's not possible to solve it?
link |
And the result on average across the kids was two minutes.
link |
Yeah, that's a bad sign, but that was a powerful sign
link |
that they need to learn to not give up so quickly.
link |
Yeah, we mentioned offline
link |
because we've been talking so much about visualization,
link |
Grant Sanderson, Three Blue 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
link |
into visual, like visually representing them,
link |
exploring them in ways that help you illuminate
link |
like the concepts.
link |
What do you think is the role of that?
link |
So he uses mostly programmatic visualization.
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, how scalable is that approach?
link |
But in general, what do you think about his approach?
link |
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 it is creating the visuals
link |
and then weaving a story with those visuals
link |
that kind of builds, like there's also,
link |
I mean, there's also drama in it.
link |
You start with a small example
link |
and then you kind of, all of a sudden there's a surprise.
link |
And it really, I mean, it makes you fall in love
link |
He does talk about that.
link |
His sense is like some of the stuff,
link |
he doesn't feel like he's teaching like the core curriculum,
link |
which is something, he sees himself
link |
as an inspirational figure.
link |
But because I think it's too difficult
link |
to kind of convert all of the curriculum into those elements.
link |
And probably you don't need to.
link |
I mean, if people get to experience mathematical ideas
link |
in the way that he shares them, 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 |
He created a library called Manim and he open sourced it.
link |
And that library is the, people should check it out.
link |
It's written in Python
link |
and it 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 super well documented
link |
or perfect, but him releasing that,
link |
now there's all of these people educating it.
link |
And the cool, to me personally,
link |
the coolest thing is to see like people they're not,
link |
you know, don't have like a million subscribers
link |
or something is 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 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,
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it shows that that impact.
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I think just giving people that idea
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that you can do that with maths 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 that people do like JavaScript or Python.
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You can build like visualizations of most concepts
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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, people will really love it.
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People actually, people love visualizations of math.
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Cause they, I mean, it's something in us
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that loves patterns, loves figuring out difficult things
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and the patterns in there then 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 I'm in now,
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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 and patterns,
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the breaking of symmetry and patterns.
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And then it's funny that we don't see mathematics
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as somehow intricately connected to that, but it is, right?
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I mean, that's one of the perspectives
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that I love students to take is to be a pattern seeker.
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In, yeah, 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, but, you know,
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when you think about multiplying by five
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and the fact you can, you know,
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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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So you ultimately think like the Udacity models
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is a little bit more effective
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than just a plain two hour lecture.
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I think there's definitely,
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you can bring in good pedagogy 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
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and make it more equal came about
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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 relationships.
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So it is true that we can do that with some words
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that aren't, it's not a huge change to the education system.
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Do you have advice for young people?
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We've been talking about mathematics quite a bit
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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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of how to live a life 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, my advice would be to always,
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it sounds really corny,
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but always believe in yourself
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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?
link |
And they always come away not being able to find it.
link |
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
link |
that have what we call special needs.
link |
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, you can learn anything
link |
is something I wish all people would have.
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Actually also applies when you've achieved
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some level of success too.
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What I find, like in my life with people that love me,
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when you achieve success,
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they keep celebrating your success
link |
and they want you to keep doing the thing
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that you were successful at,
link |
as opposed to believing in that you can do something else,
link |
something big, whatever your heart says to do.
link |
And one of the things that I realized the value of this,
link |
quite recently, which is sad to say,
link |
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.
link |
It's like a little bit is on you.
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It's like, you don't get that sometimes
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if you go to 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
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they're interested in.
link |
But the reality is the people you surround yourself with,
link |
they're going to define your life trajectory.
link |
So select people that believe in you.
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And get away from people who don't believe in you.
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Sometimes parents can be that, they can love you deeply,
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but they set, it's the math thing we mentioned,
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they might set certain constraints on the beliefs
link |
And so in that, if you're interested in mathematics,
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and 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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Yeah, 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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So sad the number of people who've had
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those negative messages from parents.
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In my Limitless Mind book, I interviewed quite a few people
link |
who'd been told they couldn't do math,
link |
sometimes by parents, sometimes by teachers.
link |
And fortunately, they had got other ideas
link |
at some point in their life,
link |
and realized there was this whole world
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of mathematical thinking that was open to them.
link |
So it's really important that people do connect
link |
with people who believe in them,
link |
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,
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100 years from now?
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Are you optimistic about this future?
link |
Yeah, I definitely have hope.
link |
There is, change can happen in the education system.
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In recent years, it's been microscopically slow.
link |
But I do actually see change happening.
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We were talking earlier that data science is now
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a course you can take in high school instead of algebra two.
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And that's pretty amazing because that content
link |
was set out in 1892 and hasn't changed since then.
link |
And so now we're actually seeing a change
link |
in the content of high school.
link |
So I'm amazed that that's happening
link |
and very happy it's happening.
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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,
link |
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 |
And it's interesting to think how certain tools
link |
like programming, maybe they'll be deeply integrated
link |
in everything we do.
link |
Yeah, you would think that all kids are growing up
link |
learning to program and create.
link |
So I just think, I mean, the system of schooling
link |
we have now, people call it a factory model.
link |
It's not designed to inspire creativity.
link |
And I feel like that will also change.
link |
People might look back on these days
link |
and think they were hilarious,
link |
but maybe we'll in the future,
link |
kids will be doing their own programming
link |
and they'll be able to learn things
link |
and find out things and create things
link |
even as they're learning.
link |
And maybe the individual subject boundaries will go.
link |
Data science itself coming into the education system
link |
kind of illustrates that because people realize
link |
it doesn't really fit inside any of the subjects.
link |
So what do we do with it?
link |
Where does it go and who teaches it?
link |
So it's already raising those kind of questions
link |
and questioning how we have these different subject boundaries.
link |
So you've seen data science be integrated
link |
into the curriculum?
link |
Yes, it's happening across the United States as we speak.
link |
I wonder how they got initiated.
link |
Like how does change happen in the education system?
link |
Is it just a few revolutionary leaders?
link |
It does, I think so.
link |
It's been an interesting journey
link |
seeing data science take off actually.
link |
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,
link |
the 50,000 high schools saying, we now accept data science.
link |
Kids can take it instead of algebra two.
link |
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.
link |
In this modern age.
link |
Then it goes viral.
link |
California's ahead, I think in creating courses
link |
and having kids go through it, but it's suddenly,
link |
when I last looked, there were 12 states
link |
that were allowing data science as a high school course.
link |
And I think by next year, that will have doubled or more.
link |
So change is happening.
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Joe, as I said, I think mathematics
link |
is truly a beautiful subject.
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And you having an impact on millions of people
link |
and you having an impact on millions of people's lives
link |
by educating them, by inspiring teachers to educate
link |
in the ways that you've talked about
link |
in multidimensional ways, in visual ways,
link |
I think is incredible.
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So you're spreading beauty into the world.
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So I really, really appreciate
link |
that you spend 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 Boller.
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To support this podcast,
link |
please check out our sponsors in the description.
link |
And now let me leave you with some words
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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.