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. (Laughter) 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, it's just a losing proposition.
我想请你回忆过去, 当你真的热爱过一些东西, 一部电影,一张专辑,一首歌曲或一本书, 并且你全心全意地推荐它 给你很喜欢的某个人, 你期望他的反应,你一直等待着, 但反馈回来的却是他很讨厌它。 因此,以这种方式介绍, 在这种完全相同的状态下, 我度过了在过去六年工作中的每一天。 我教授高中数学。 我是将一种产品卖到一个 不受欢迎的市场,但是依据法律强制购买它。 我的意思是,这是一种 - 它只是一个亏本生意。
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.
因此,关于学数学的学生和你们所有人,我看到一个很形象的 刻板印象。 我可以给你们 一个代数Ⅱ的期末考试, 我不会期望 超过百分之25的合格率。 这两个事实并不表示你们的问题或是我学生的问题, 它表现的是在当今的美国,我们所谓的数学教育 所面临的问题。
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'd like 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.
首先,我想把数学分成两类。 一个是计算。这是你们已经忘记的东西。 例如,首项系数大于1的 保理二次方程式。 这东西也很容易重新学习, 只要你有一个关于推理,数学推理真正过硬 的基础。 我们将这称之为 我们周围世界里的数学程序应用。 这点是很难教授的。 这也是我们希望学生能够掌握并应用, 即使他们不进入数学领域。 这也是在美国,由于我们教授它的方式 以致于学生不会应用它。 所以,我要谈谈这是为什么, 为什么这是当今社会的灾难,我们可以为此做什么, 而且,更进一步,为什么现在成为一名数学老师, 这正是一个千载难逢的好时代。
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 is eagerly looking for the formula to apply in that situation. This is really destructive.
首先,你在数学课堂上 做的五个征兆 是在数学推理上错误的。 一个是缺乏主动性,你的学生没有自我行动。 你完成你的演讲, 然后马上,你就会看到五个人举手 要求你到他们课桌前重新解释整件事。 二学生缺乏坚持不懈。 三他们缺少持续记忆力,你会发现自己 在3个月后需要完完全全的重新解释这些概念。 四还存在着一种对文字题的反感, 这占了我学生中的99%。 五这种情况下,剩下的百分之一 正在急切地寻找公式 来解决问题。 这实在是毁灭性的。
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.
大卫·米尔希是“死木”美剧以及其他的精彩电视节目的创造者, 对此他有一个很好的描述。 他发誓要停止创造 当代戏剧, 及当今的单元连续剧, 因为他意识到,当人们的脑中 一天四小时想到,例如,“两个半男人”,没有冒犯的意思, 这就形成了神经传导通路,他说, 以这样一种方式,人们期望简单的问题。 他称这是“一种无法做决定的焦虑。” 人们对不能快速解决的事情不耐烦。 人们期望情景喜剧一般可以在22分中结束, 3个广告中断和一个笑声背景。 我告诉你们所有人, 如大家知道的,解决任何问题都不是那样简单。 我对此很担忧, 因为我将要在我的学生主宰的世界中退休。 当我以这种方式教学,我正在做 对我自己的将来和幸福感 不利的事情。 我在这里告诉大家,我们的教科书,特别是 大众常用的教科书,它们教授数学理论 和耐心解决的数学问题, 功能上和打开“好汉两个半”电视系列剧一样,然后这就结束了(消磨时间)。
(Laughter)
(笑声)
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; where you didn't have a surplus of information and you had to filter it out, or you didn't have sufficient information and 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 actual 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.
非常严肃地说,这是物理教科书上的一个例子, 它同样适用于数学。 首先注意到这里 大家有正好3个信息, 每一个都可以表示成某个公式, 最终, 学生可以以此计算。 我相信在真实生活中。 问问自己,什么问题你们解决过的, 值得解决的, 在此问题中,你提前知道所有信息, 或者你没有多余信息,然后你需要过滤信息, 或是你没有足够的信息, 你需要去查找些信息。 我肯定大家都同意解决问题都不是如上所述那么简单。 我认为,教科书实在是误人子弟。 因为,看看这个,这是一个练习题, 当要解决实际问题时, 我们在这就有类似问题, 我们只是在交换数字和稍稍调整内容。 如果学生们不知道这类数学题模式, 而它会帮助向你解释例题, 什么样题你可以返回去找出公式。 你确实可以,我很认真的说, 通过这个测试,没有任何物理知识, 仅仅知道如何解译教科书。那是一种耻辱。
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.
我能在数学中更准确得解决这个问题。 这是一个很酷的问题。我喜欢这个。 它是关于确定陡度和坡度 在一个滑雪缆车上. 但在这里有四个不同层面。 我很好奇你们哪个可以看到这四个不同层面的, 特别是,它们如何压缩在一起, 一次性的展现给学生, 又如何轻易地解决了问题。 我将在这里详细说明它们。你有图像。 你也有数学结构, 谈到格子,测量,标注, 点,轴,诸如此类。 你有子步骤,它们都引到我们真正想谈论的问题, 哪段是最陡的?
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 to put to you that 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.
我希望你们能看到。 我真的希望你们能看到,我们在这里 是在谈一个激发兴趣的问题,一个有意思的答案, 但我们在开拓一个平整的,简单明了的路径, 从一点到另一点, 然后祝贺我们的学生,为他们 能够很好的沿路踏过小缝隙。 这就是我们做的。 所以我想展示给你们的是,如果我们能以不同方式分开这些, 和学生们一起建造它们, 我们可以迎刃处理所有需耐心解决的问题。
So right here I start with the 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 would 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.
就在这里,我以图像开始, 我马上问个问题: 哪个部分最陡? 这会引起交谈, 因为这个图像创造的方式使你可以用两个答案辩解。 这样你可以促进人们互相争论, 朋友对朋友的争论, 一对对,日志式,随便怎样。 然后我们最终认识到 谈到 在屏幕左手边滑雪的人 或是刚上中线的滑雪的人,这是挺讨厌的。 我们认识到如果 我们更简单地谈论带有一些a,b,c,d的标志, 这该是多么好。 然后我们确定陡峭程度是指什么, 我们认识到这是一个好主意,如果有一些测量 来真的缩小范围,特别是那意味着什么。 然后,在此之后, 我们放下那个数学架构。 数学服务于对话, 对话却不为数学服务。 在这一点,我将展示给你们,10个班里有9个班 能很好的继续解决整个坡度,陡度的问题。 如果你需要 你的学生可以在一起展开那些子步骤。
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.
你们可以比较这边这个和那个-- 哪个班处理了耐心解决的问题,以及数学推理呢? 在我的实践中,我一目了然。 我这就要谈谈爱因斯坦, 我认为他做了应有的贡献。 他谈到规划问题的能力是如此重要, 但在我的实践中, 在美国这儿, 我们仅仅把问题交给学生; 我们不把他们参与到问题的形成和规划过程中。
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? 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 side 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.
所以我做的90%的事 和我每周5小时的准备时间 是拿问题中非常吸引人的元素, 像在我的教科书里的这些问题, 把它们重组成能支持数学推理和耐心解决问题的方法。 这里是它如何运作的。 我喜欢这个问题,这是关于一个水桶。 问题是:多长时间可以装满这个水桶呢? 第一件事,我们简化子步骤。 学生们必须得展开这些子步骤, 用公式化表达它们。 然后认识到写在那里的东西是他们所需要的所有信息。 没有一个是多余信息,用不着删除。 学生们需要决定,好了, 高度重要吗?它的尺码重要吗? 开关的颜色重要吗?这里什么是重要的? 这在数学课里是不具代表性的问题。 这样在这里我们有了一个水桶。 多长时间会把它装满,就是这样。
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 then 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?
因为这是21世纪, 我们喜欢以真实世界的表达方式来谈论真实的世界, 不是以你常在教科书上看到的线条艺术 或是插图艺术来表达。 我们出去应用,并给它照个相。 现在我们有了真实的东西。 它要多长时间装满? 更好的,我们录像, 一个人在装满它的录像。 它在慢慢的装满,令人不耐烦的等待。 这很乏味。 学生们在看他们的表,翻白眼, 他们时不时地想, “伙计,这个要多长时间装满呐?” (笑声) 这就是你想要引起关注的地方,好。
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, every student 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, 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.
这个问题,就在这里,对我来说很有趣。 因为,像个引言, 我教小孩子,因为我的经验, 我教那些需要辅导的孩子。 我不教那些不参与数学对话的孩子, 因为他们懂公式, 也知道如何更好地运用公式,甚至比我好。 所以我不讨论这个。 但在这里,每个人都在同一个水平上运用直觉。 每个人都装过水, 所以我让孩子们自己回答这个问题:多长时间装满水。 我让这些在数学上和交谈上感到恐惧的孩子们 也加入谈论。 我们把名字写到黑板上,把它们连到猜测的答案上, 孩子们把答案带到这里。 然后孩子们遵照我描述的过程。 最好的部分,或者说,一个更好的部分 是我们没有从 教师版教科书的后面得到答案。 我们却看到了录像的结尾。 (笑声) 这很有意思,对吧。 因为理论模型总是得出 和教师版后面的答案一样的答案, 这很好, 但谈到错误的来源, 当理论不符合实际的时候,就让人害怕。 但那些谈论是如此珍贵, 是最有价值的。
So I'm here to report some really fun games 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.
所以我在这里报导一些很有趣的收获, 关于在数学课程刚开始的时候,学生带有这些预置的 错误征兆。 这些孩子现在已经学了一个学期, 我能在黑板上写下一些 全新的,完全陌生的问题, 他们能就此展开讨论, 这比他们在学年开始的时候长3,4分钟, 这真有意思。 孩子们不再反感数学问答题, 因为他们从新定义数学问答题。 孩子们不再恐惧数学, 因为他们在慢慢从新定义数学。 这个过程很有意思。
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.
我鼓励数学教师多多应用多媒体, 因为它把真实世界以高清晰度的图片和全色彩图像带到 你的教室, 来鼓励学生对那个场景的直觉, 来问你可能问的最短的问题, 让那些更细,更具体的问题随着交谈显露出来, 让学生们着手解决问题, 正如爱因斯坦所说, 总而言之,就是给予更少的帮助, 因为教科书在以非常错误的方式帮助大家。 它在免去大家 对耐心问题的解决和数学推理的责任,更没有帮助性。
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 line 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?
那为什么现在是作为一个数学教师的绝好时候, 这是因为我们有工具来创造 这个高质量的数学课程,就在我们的口袋里的工具手机等等。 它无所不在,非常便宜。 还有自由散播它的工具, 在公开授权下, 从没有比现在更便宜或更普遍的社交互联网工具。 不久之前,我把一系列的录像放在我的博客里, 在两周之内它得到6000的观看率。 我还收到很多我从来没有去过的国家的邮件,有的教师写邮件说: “哦,是的。我们就此展开了很好的讨论。 哦,顺便说一下,这是我如何改善你教学材料的例子。” 这么多,喔。 我最近在我的博客上提出这个问题。 在一个杂物店,你排那个队, 一个是有一个手推车和19个商品 或是另一个有4个手推车和3,5,2 和1个商品。 这里牵涉到的线性模型,它成为了我的教学中很好的材料, 但它最后让我上了在几周之前“早安,美国”的节目, 这的确是不可思议,对吧。
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 really 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. (Applause)
从所有这些,我只能下此结论 人们,不仅仅是学生, 都非常渴望这个。 数学解读世界。 数学也是描述人们 直觉的词汇。 所以我只是鼓励大家,无论你们是什么教育程度, 无论你们是学生,还是家长,老师,决策者,不论是什么, 请坚持更好地应用数学教程。 我们需要更多耐心的问题解决者。谢谢。