Teach Kids Math With Model Method
Dan Meyer: Math class needs a makeover
Today's math curriculum is teaching students to expect -- and excel at -- paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. At TEDxNYED, Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.
Dan Meyer is exploring the way we teach teachers to teach kids.
Can I ask you to please recall a time when you really loved something, a movie, an album, a song or a book, and you recommended it wholeheartedly to someone you also really liked, and you anticipated that reaction, you waited for it, and it came back, and the person hated it. So, by way of introduction, that is the exact same state in which I spent every working day of the last six years. I teach high school math. I sell a product to a market that doesn't want it, but is forced by law to buy it. I mean, that's kind of -- it's just a losing proposition.
So there's a useful stereotype about students that I see, a useful stereotype about you all. I could give you guys an algebra-two final exam, and I would expect no higher than a 25 percent pass rate. And both of these facts say less about you or my students than they do about what we call math education in the U.S. today.
To start with, I'd like to break math down into two categories. One is computation. This is the stuff you've forgotten. For example, factoring quadratics with leading coefficients greater than one. This stuff is also really easy to relearn, provided you have a really strong grounding in reasoning, math reasoning. We'll call it the application of math processes to the world around us. This is hard to teach. This is what we would love students to retain, even if they don't go into mathematical fields. This is also something that, the way we teach it in the U.S. all but ensures they won't retain it. So, I'm going to talk about why that is, why that's such a calamity for society, what we can do about it, and, to close with, why this is an amazing time to be a math teacher.
So first, five symptoms that you're doing math reasoning wrong in your classroom. One is a lack of initiative; your students don't self-start. You finish your lecture block and immediately you have five hands going up asking you to re-explain the entire thing at their desks. Students lack perseverance. They lack retention; you find yourself re-explaining concepts three months later, wholesale. There's an aversion to word problems, which describes 99 percent of my students. And then the other one percent are eagerly looking for the formula to apply in that situation. This is really destructive.
David Milch, creator of "Deadwood" and other amazing TV shows, has a really good description for this. He swore off creating contemporary drama, shows set in the present day, because he saw that when people fill their mind with four hours a day of, for example, "Two and a Half Men," no disrespect, it shapes the neural pathways, he said, in such a way that they expect simple problems. He called it, "an impatience with irresolution." You're impatient with things that don't resolve quickly. You expect sitcom-sized problems that wrap up in 22 minutes, three commercial breaks and a laugh track. And I'll put it to all of you, what you already know, that no problem worth solving is that simple. I am very concerned about this, because I'm going to retire in a world that my students will run. I'm doing bad things to my own future and well-being when I teach this way. I'm here to tell you that the way our textbooks, particularly, mass-adopted textbooks, teach math reasoning and patient problem solving, it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
In all seriousness, here's an example from a physics textbook. It applies equally to math. Notice first of all here that you have exactly three pieces of information there, each of which will figure into a formula somewhere, eventually, which the student will then compute. I believe in real life. And ask yourself, what problem have you solved, ever, that was worth solving, where you knew all of the given information in advance, or where you didn't have a surplus of information and you had to filter it out, or where you didn't have sufficient information and you had to go find some. I'm sure we all agree that no problem worth solving is like that. And the textbook, I think, knows how it's hamstringing students. Because, watch this, this is the practice problem set. When it comes time to do the actually problem set, we have problems like this right here where we're just swapping out numbers and tweaking the context a little bit. And if the student still doesn't recognize the stamp this was molded from, it helpfully explains to you what sample problem you can return to to find the formula. You could literally, I mean this, pass this particular unit without knowing any physics, just knowing how to decode a textbook. That's a shame.
So I can diagnose the problem a little more specifically in math. Here's a really cool problem. I like this. It's about defining steepness and slope using a ski lift. But what you have here is actually four separate layers. And I'm curious which of you can see the four separate layers, and, particularly, how when they're compressed together and presented to the student all at once, how that creates this impatient problem solving. I'll define them here. You have the visual. You also have the mathematical structure, talking about grids, measurements, labels, points, axes, that sort of thing. You have substeps, which all lead to what we really want to talk about, which section is the steepest.
So I hope you can see. I really hope you can see how, what we're doing here is taking a compelling question, a compelling answer, but we're paving a smooth, straight path from one to the other, and congratulating our students for how well they can step over the small cracks in the way. That's all we're doing here. So I want put to you, if we can separate these in a different way and build them up with students, we can have everything we're looking for in terms of patient problem solving.
So right here, I start with a visual, and I immediately ask the question: Which section is the steepest? And this starts conversation because the visual is created in such a way where you can defend two answers. So you get people arguing against each other, friend versus friend, in pairs, journaling, whatever. And then eventually we realize it's getting annoying to talk about the skier in the lower left-hand side of the screen or the skier just above the mid line. And we realize how great would it be if we just had some A, B, C, and D labels to talk about them more easily. And then as we start to define what does steepness mean, we realize it'd be nice to have some measurements to really narrow it down, specifically what that means. And then and only then, we throw down that mathematical structure. The math serves the conversation. The conversation doesn't serve the math. And at that point, I'll put it to you that nine out of 10 classes are good to go on the whole slope, steepness thing. But if you need to, your students can then develop those substeps together.
Do you guys see how this, right here, compared to that -- which one creates that patient problem solving, that math reasoning? It's been obvious in my practice, to me. And I'll yield the floor here for a second to Einstein, who, I believe, has paid his dues. He talked about the formulation of a problem being so incredibly important, and yet in my practice, in the U.S. here, we just give problems to students; we don't involve them in the formulation of the problem.
So 90 percent of what I do with my five hours of prep time per week is to take fairly compelling elements of problems like this from my textbook and rebuild them in a way that supports math reasoning and patient problem solving. And here's how it works. I like this question. It's about a water tank. The question is: How long will it take you to fill it up? Okay? First things first, we eliminate all the substeps. Students have to develop those. They have to formulate those. And then notice that all the information written on there is stuff you'll need. None of it's a distractor, so we lose that. Students need to decide, all right, well, does the height matter? Does the size of it matter? Does the color of the valve matter? What matters here? Such an underrepresented question in math curriculum. So now we have a water tank. How long will it take you to fill it up, and that's it.
And because this is the 21st century, and we would love to talk about the real world on its own terms, not in terms of line art or clip art that you so often see in textbooks, we go out, and we take a picture of it. So now we have the real deal. How long will it take it to fill it up? And, even better, is we take a video, a video of someone filling it up. And it's filling up slowly, agonizingly slowly. It's tedious. Students are looking at their watches, rolling their eyes, and they're all wondering at some point or another, "Man, how long is it going to take to fill up?" (Laughter) That's how you know you've baited the hook, right.
And that question, off this right here, is really fun for me, because, like the intro, I teach kids, because of my inexperience, I teach the kids that are the most remedial, all right. And I've got kids who will not join a conversation about math because someone else has the formula, someone else knows how to work the formula better than me. So I won't talk about it. But here, everyone is on a level playing field of intuition. Everyone's filled something up with water before, so I get kids answering the question, how long will it take. I've got kids who are mathematically and conversationally intimidated joining the conversation. We put names on the board, attach them to guesses, and kids have bought in here. And then we follow the process I've described. And the best part here, or one of the better parts is that we don't get our answer from the answer key in the back of the teacher's edition. We, instead, just watch the end of the movie. (Laughter) And that's terrifying, all right. Because the theoretical models that always work out in the answer key in the back of a teacher's edition, that's great, but it's scary to talk about sources of error when the theoretical does not match up with the practical. But those conversations have been so valuable, among the most valuable.
So I'm here to report some really fun gains with students who come pre-installed with these viruses day one of the class. These are the kids who now, one semester in, I can put something on the board, totally new, totally foreign, and they'll have a conversation about it for three or four minutes more than they would have at the start of the year, which is just so fun. We're no longer averse to word problems, because we've redefined what a word problem is. We're no longer intimidated by math, because we're slowly redefining what math is. This has been a lot of fun.
I encourage math teachers I talk to to use multimedia, because it brings the real world into your classroom in high resolution and full color, to encourage student intuition for that level playing field, to ask the shortest question you possibly can and let those more specific questions come out in conversation, to let students build the problem, because Einstein said so, and to finally, in total, just be less helpful, because the textbook is helping you in all the wrong ways. It's buying you out of your obligation for patient problem solving and math reasoning, to be less helpful.
And why this is an amazing time to be a math teacher right now is because we have the tools to create this high-quality curriculum in our front pocket. It's ubiquitous and fairly cheap. And the tools to distribute it freely, under open licenses has also never been cheaper or more ubiquitous. I put a video series on my blog not so long ago, and it got 6,000 views in two weeks. I get emails still from teachers in countries I've never visited saying, "Wow, yeah. We had a good conversation about that. Oh, and by the way, here's how I made your stuff better," which, wow. I put this problem on my blog recently. In a grocery store, which line do you get into, the one that has one cart and 19 items or the one with four carts and three, five, two and one items. And the linear modeling involved in that was some good stuff for my classroom, but it eventually got me on "Good Morning America" a few weeks later, which is just bizarre, right.
And from all of this, I can only conclude that people, not just students, are really hungry for this. Math makes sense of the world. Math is the vocabulary for your own intuition. So I just encourage you, whatever your stake is in education, whether you're a student, parent, teacher, policy maker, whatever, insist on better math curriculum. We need more patient problem solvers. Thank you.
Sir Ken Robinson: Bring on the learning revolution!
In this poignant, funny follow-up to his fabled 2006 talk, Sir Ken Robinson makes the case for a radical shift from standardized schools to personalized learning -- creating conditions where kids' natural talents can flourish
Creativity expert Sir Ken Robinson challenges the way we're educating our children. He champions a radical rethink of our school systems, to cultivate creativity and acknowledge multiple types of…
I was here four years ago, and I remember, at the time, that the talks weren't put online; I think they were given to TEDsters in a box, a box set of DVDs, which they put on their shelves, where they are now.
And actually Chris called me a week after I'd given my talk and he said, "We're going to start putting them online. Can we put yours online?" And I said, "Sure."
And four years later, as I said, it's been seen by four ... Well, it's been downloaded four million times. So I suppose you could multiply that by 20 or something to get the number of people who've seen it. And as Chris says, there is a hunger for videos of me.
... don't you feel?
So, this whole event has been an elaborate build-up to me doing another one for you, so here it is.
Al Gore spoke at the TED Conference I spoke at four years ago and talked about the climate crisis. And I referenced that at the end of my last talk. So I want to pick up from there because I only had 18 minutes, frankly. So, as I was saying...
You see, he's right. I mean, there is a major climate crisis, obviously. And I think if people don't believe it, they should get out more. (Laughter) But I believe there's a second climate crisis, which is as severe, which has the same origins, and that we have to deal with with the same urgency. And I mean by this -- and you may say, by the way, "Look, I'm good. I have one climate crisis; I don't really need the second one." But this is a crisis of, not natural resources, though I believe that's true, but a crisis of human resources.
I believe, fundamentally, as many speakers have said during the past few days, that we make very poor use of our talents. Very many people go through their whole lives having no real sense of what their talents may be, or if they have any to speak of. I meet all kinds of people who don't think they're really good at anything.
Actually, I kind of divide the world into two groups now. Jeremy Bentham, the great utilitarian philosopher, once spiked this argument. He said, "There are two types of people in this world, those who divide the world into two types and those who do not." (Laughter) Well, I do. (Laughter)
I meet all kinds of people who don't enjoy what they do. They simply go through their lives getting on with it. They get no great pleasure from what they do. They endure it, rather than enjoy it, and wait for the weekend. But I also meet people who love what they do and couldn't imagine doing anything else. If you said to them, "Don't do this anymore," they'd wonder what you were talking about. Because it isn't what they do, it's who they are. They say, "But this is me, you know. It would be foolish for me to abandon this, because it speaks to my most authentic self." And it's not true of enough people. In fact, on the contrary, I think it's certainly a minority of people. And I think there are many
possible explanations for it. And high among them is education, because education, in a way, dislocates very many people from their natural talents. And human resources are like natural resources; they're often buried deep. You have to go looking for them. They're not just lying around on the surface. You have to create the circumstances where they show themselves. And you might imagine education would be the way that happens. But too often, it's not. Every education system in the world is being reformed at the moment. And it's not enough. Reform is no use anymore, because that's simply improving a broken model. What we need -- and the word's been used many times during the course of the past few days -- is not evolution, but a revolution in education. This has to be transformed into something else.
One of the real challenges is to innovate fundamentally in education. Innovation is hard because it means doing something that people don't find very easy for the most part. It means challenging what we take for granted, things that we think are obvious. The great problem for reform or transformation is the tyranny of common sense -- things that people think, "Well, it can't be done any other way because that's the way it's done."
I came across a great quote recently from Abraham Lincoln, who I thought you'd be pleased to have quoted at this point. (Laughter) He said this in December 1862 to the second annual meeting of Congress. I ought to explain that I have no idea what was happening at the time. We don't teach American history in Britain. (Laughter) We suppress it. You know, this is our policy. (Laughter) So, no doubt, something fascinating was happening in December 1862, which the Americans among us will be aware of.
But he said this: "The dogmas of the quiet past are inadequate to the stormy present. The occasion is piled high with difficulty, and we must rise with the occasion." I love that. Not rise to it, rise with it. "As our case is new, so we must think anew and act anew We must disenthrall ourselves and then we shall save our country."
I love that word, "disenthrall." You know what it means? That there are ideas that all of us are enthralled to, which we simply take for granted as the natural order of things, the way things are. And many of our ideas have been formed, not to meet the circumstances of this century, but to cope with the circumstances of previous centuries. But our minds are still hypnotized by them. And we have to disenthrall ourselves of some of them. Now, doing this is easier said than done. It's very hard to know, by the way, what it is you take for granted. And the reason is that you take it for granted.
So let me ask you something that you may take for granted. How many of you here are over the age of 25? That's not what I think you take for granted. I'm sure you're familiar with that already. Are there any people here under the age of 25? Great. Now, those over 25, could you put your hands up if you're wearing a wristwatch? Now that's a great deal of us, isn't it? Ask a room full of teenagers the same thing. Teenagers do not wear wristwatches. I don't mean they can't or they're not allowed to, they just often choose not to. And the reason is, you see, that we were brought up in a pre-digital culture, those of us over 25. And so for us, if you want to know the time, you have to wear something to tell it. Kids now live in a world which is digitized, and the time, for them, is everywhere. They see no reason to do this. And, by the way, you don't need to do it either; it's just that you've always done it, and you carry on doing it. My daughter never wears a watch, my daughter Kate, who's 20. She doesn't see the point. As she says, "It's a single function device." (Laughter) "Like, how lame is that?" And I say, "No, no, it tells the date as well." (Laughter) "It has multiple functions."
But you see, there are things we're enthralled to in education. Let me give you a couple of examples. One of them is the idea of linearity, that it starts here, and you go through a track, and if you do everything right, you will end up set for the rest of your life. Everybody who's spoken at TED has told us implicitly, or sometimes explicitly, a different story, that life is not linear, it's organic. We create our lives symbiotically as we explore our talents in relation to the circumstances they help to create for us. But you know, we have become obsessed with this linear narrative. And probably the pinnacle for education is getting into college. I think we are obsessed with getting people to college, certain sorts of college. I don't mean you shouldn't go to college, but not everybody needs to go, and not everybody needs to go now. Maybe they go later, not right away.
And I was up in San Francisco a while ago doing a book signing. There was this guy buying a book, he was in his 30s. And I said, "What do you do?" And he said, "I'm a fireman." And I said, "How long have you been a fireman?" He said, "Always, I've always been a fireman." And I said, "Well, when did you decide?" He said, "As a kid." He said, "Actually, it was a problem for me at school, because at school, everybody wanted to be a fireman." He said, "But I wanted to be a fireman." And he said, "When I got to the senior year of school, my teachers didn't take it seriously. This one teacher didn't take it seriously. He said I was throwing my life away if that's all I chose to do with it, that I should go to college, I should become a professional person, that I had great potential, and I was wasting my talent to do that." And he said, "It was humiliating because he said it in front of the whole class, and I really felt dreadful. But it's what I wanted, and as soon as I left school, I applied to the fire service and I was accepted." And he said, "You know, I was thinking about that guy recently, just a few minutes ago when you were speaking, about this teacher," he said, "because six months ago, I saved his life." (Laughter) He said, "He was in a car wreck, and I pulled him out, gave him CPR, and I saved his wife's life as well." He said, "I think he thinks better of me now."
You know, to me, human communities depend upon a diversity of talent, not a singular conception of ability. And at the heart of our challenges -- (Applause) At the heart of the challenge is to reconstitute our sense of ability and of intelligence. This linearity thing is a problem.
When I arrived in L.A. about nine years ago, I came across a policy statement, very well-intentioned, which said, "College begins in kindergarten." No, it doesn't. (Laughter) It doesn't. If we had time, I could go into this, but we don't. (Laughter) Kindergarten begins in kindergarten. (Laughter) A friend of mine once said, "You know, a three year-old is not half a six year-old." (Laughter) (Applause) They're three.
But as we just heard in this last session, there's such competition now to get to kindergarten, to get to the right kindergarten, that people are being interviewed for it at three. Kids sitting in front of unimpressed panels, you know, with their resumes, (Laughter) flipping through and saying, "Well, this is it?" (Laughter) (Applause) "You've been around for 36 months, and this is it?" (Laughter) "You've achieved nothing, commit. Spent the first six months breastfeeding, the way I can see it." (Laughter) See, it's outrageous as a conception, but it attracts people.
The other big issue is conformity. We have built our education systems on the model of fast food. This is something Jamie Oliver talked about the other day. You know there are two models of quality assurance in catering. One is fast food, where everything is standardized. The other are things like Zagat and Michelin restaurants, where everything is not standardized, they're customized to local circumstances. And we have sold ourselves into a fast food model of education. And it's impoverishing our spirit and our energies as much as fast food is depleting our physical bodies.
I think we have to recognize a couple of things here. One is that human talent is tremendously diverse. People have very different aptitudes. I worked out recently that I was given a guitar as a kid at about the same time Eric Clapton got his first guitar. You know, it worked out for Eric, that's all I'm saying. (Laughter) In a way, it did not for me. I could not get this thing to work no matter how often or how hard I blew into it. It just wouldn't work.
But it's not only about that. It's about passion. Often, people are good at things they don't really care for. It's about passion, and what excites our spirit and our energy. And if you're doing the thing that you love to do, that you're good at, time takes a different course entirely. My wife's just finished writing a novel, and I think it's a great book, but she disappears for hours on end. You know this, if you're doing something you love, an hour feels like five minutes. If you're doing something that doesn't resonate with your spirit, five minutes feels like an hour. And the reason so many people are opting out of education is because it doesn't feed their spirit, it doesn't feed their energy or their passion.
So I think we have to change metaphors. We have to go from what is essentially an industrial model of education, a manufacturing model, which is based on linearity and conformity and batching people. We have to move to a model that is based more on principles of agriculture. We have to recognize that human flourishing is not a mechanical process, it's an organic process. And you cannot predict the outcome of human development; all you can do, like a farmer, is create the conditions under which they will begin to flourish.
So when we look at reforming education and transforming it, it isn't like cloning a system. There are great ones like KIPPs, it's a great system. There are many great models. It's about customizing to your circumstances, and personalizing education to the people you're actually teaching. And doing that, I think is the answer to the future because it's not about scaling a new solution; it's about creating a movement in education in which people develop their own solutions, but with external support based on a personalized curriculum.
Now, in this room, there are people who represent extraordinary resources in business, in multimedia, in the internet. These technologies, combined with the extraordinary talents of teachers, provide an opportunity to revolutionize education. And I urge you to get involved in it because it's vital, not just to ourselves, but to the future of our children. but we have to change from the industrial model to an agricultural model, where each school can be flourishing tomorrow. That's where children experience life. Or at home, if that's where they choose to be educated with their families or their friends.
There's been a lot of talk about dreams over the course of this few days. And I wanted to just very quickly -- I was very struck by Natalie Merchant's songs last night, recovering old poems. I wanted to read you a quick, very short poem from W.B. Yeats, who's someone you may know. He wrote this to his love, Maud Gonne, and he was bewailing the fact that he couldn't really give her what he thought she wanted from him. And he says, "I've got something else, but it may not be for you."
He says this: "Had I the heavens embroidered cloths, Enwrought with gold and silver light, The blue and the dim and the dark cloths of night and light and the half-light, I would spread the cloths under your feet; But I, being poor, have only my dreams; I have spread my dreams under your feet; Tread softly because you tread on my dreams." And every day, everywhere, our children spread their dreams beneath our feet. And we should tread softly.
Thank you very much.
New iPhone/iPad App Game Brings Together Puzzles and Math
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