TED日本語 - ダン・メイヤー: 数学クラス改造計画



TED日本語 - ダン・メイヤー: 数学クラス改造計画

TED Talks

Math class needs a makeover
Dan Meyer




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.

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'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.

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.

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; 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.

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 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.

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.

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? 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.

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?

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.

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.

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 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?

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)

大好きなものを 思い出して下さい 映画 アルバム 歌 本など それを好きな人に 本気で勧めたとします 相手の反応を期待して待ちますが 返事は正反対、という結末 前置きなのですが この状態こそまさしく 6年間毎日経験した事です 私は高校数学教師です 要らない人に 法律なので、とモノを売るのです 失敗元々の商売です

説明に便利な生徒の典型例があります 皆さんにも当てはまるかもしれません 代数IIの 期末試験を行うとします 合格率は 25%以下だと思います これは皆さんや生徒の事より アメリカの数学教育の現状を 物語っています

まず数学を2つに分類します 1つは計算。お忘れだと思います 例えば、2次方程式の因数分解です 1より大きい首位係数のものです この手の問題は 数的推論の基本が出来ていれば すぐ思い出せます 世界に対する 数学的プロセスの適用です 教えるのは難しいですが 数学を志すかどうかに関わらず 生徒に身につけて欲しい項目です 現在のアメリカの教え方では まず身に付かないでしょう 今日のテーマは そんな教育に対して何ができるか、です なぜ現代が数学教師にとって 驚くべき時代なのかにも触れます

学校で間違った方法で 数的推理が教えられて兆候が 5つあります 1つは率先力欠如で、自ら学習しません 説明が終わるとすぐに 生徒5人が手を挙げ また全て説明してくれと言うのです 忍耐力も欠けています 記憶力もありません 3ヵ月後にはまた全部説明です 文章題への嫌悪感もあります 生徒の99%はそうです 残りの1%は 公式が当てはまらないかと 一生懸命に探っています 目も当てられない状況です

「デッドウッド」の創作者のデイビッド ミルチ氏は これを上手く説明します 彼は 現代を舞台にした 現代劇の創作を止めました 「チャーリーシーンのハーパーボーイズ」を 毎日4時間も見ていると 特殊な神経経路ができ 単純問題を期待するようになるというのです 彼の言う「決断できないじれったさ」です すぐ解決しないものに我慢できないのです 22分で問題解決するホームコメディー式を望みます CM3回と笑い声付きです このように話していますが 解決価値のある問題は単純ではありません 心配しているのは 将来生徒の指揮下で暮らすのですよ 自分自身の未来と福祉に 役に立たないことをしているのです 教えながら 現在の教科書、特にー 全国版の教科書で数的推論と 問題解決を教えるなら 「ハーパーボーイズ」を観るのと同じです


さてこれが例です。物理の教科書です 数学に当てはまります まず見て頂きたいのは この3つの情報で この1つ1つがやがて公式となり 最終的には生徒が それを使って計算します 現実生活では違います 解決価値のある問題は どんなものを解決したことがありますか 情報を全て入手している場合ですよ 情報が余分にあれば取捨選択したでしょう 情報が不十分なら 自ら探しに行ったはずです 有価値の問題でそのようなものは皆無です 教科書も生徒の思考力を阻害しています 理由はこれをご覧下さい。練習問題です 問題に取り組む段になると こういう問題が控えています 数字と内容を少しいじっただけです それでも出題元がわからない場合は 公式が載っている例題を 親切に教えてくれます そうやって1つの単元を 文字通り通過します。物理は全く学ばず 教科書のどこに何があるかだけを学びます

もう少し具体的に数学で検討しましょう ここに好都合な問題があります スキーリフトを使って 傾斜角とスロープの長さを求めます 4段階の手順があります どなたかその4つの手順がわかりますか 生徒には凝縮して 一度に提示されるのですよ それでじれったい問題解決が起こります 実際に解いてみましょう。図があります 皆さんは数学の基礎知識もお持ちです 格子、長さ、名称 点、軸などのことです また分割して考える手順もご存知です どの部分が最も険しいですか

話が見えてきたと思います 私達は誘導的な問いと 解答という形をとっています しかし問いから問いへと まっすぐな道を整備してやり 途中の小さな割れ目をうまく 踏み越えれば褒めてやります ただそういうことなのです もし違うやり方でこれらを分解して 生徒と一緒に組み立てるなら まさに問題解決の理想形になるのです

この図から始めましょう すぐに質問をします どの部分が最も険しいですか こう切り出すのは 解答が2つあるようになっているからです こうして互いに議論させます 友人同士で ペアになって記録もとります そのうち気づきます スキーヤーは目障りなだけだと 画面の左下あるいは 真ん中より少し上の絵です またこう感じます A~Dに名称があれば 話が簡単になるのにと 勾配の意味を定義し始めるとき その基準となる数値が あればいいと気づきます そのとき初めて 数学的思考法を投げ与えます 数学は議論に役立ちますが 議論は数学に役立ちません その時点で10クラスの内9クラスが 問題への準備ができています 必要なら生徒自身が 小ステップを考えることもできます

これが辛抱強い問題解決 数的推論を生み出すのです 私の実践では明白でした ここでアインシュタイン君に発言権です これは当然ですよね 彼は問題の定式化が肝要だと言います でも米国の私の授業では ただ生徒に問題を与えるだけです 生徒は問題の定式化に関わりません

週5時間の準備時間の 90%を使い かなり誘導的要素をもつ問題を 教科書から選んで 数的推論と問題解決の練習用に再編します それはこういうことです 水タンクの良問です 満たすのにどれだけ時間がかかりますか 真っ先に小ステップを排除します 生徒が自分で考え 定式化しなければなりません そこにある全情報が必要と気づくでしょう どれも解答の邪魔になりません ここで生徒が考えますー 高さや大きさは大事かな? バルブの色は?ここで何が重要でしょう 教科課程では提示されていません ここに水タンクがあります 満たすのに時間は?それだけです

21世紀である現在 私達は現実世界をありのままに見ます それは教科書でよく目にする線アートや クリップアートではありません 外に出て、写真を撮ります 現実を扱っているのです 満たすのにどれだけかかりますか ビデオを撮ればもっといいでしょう 誰かがタンクを満たしているところです 腹立たしいほどゆっくりで 退屈です 生徒は時計を見ています ある時点でこう思います 「おい、一体どれだけかかるんだよ」 (笑) こうして問題に食いついたのです

ここからが本当に面白いのです 冒頭で言いましたように 私は経験が浅いので 最も遅れている子供に教えています 数学の話には参加しない子供もいます 誰かが公式を知っており 私よりその使い方を知っているからです だからそれについて話しません でもここでは皆同じ直観力を持っています 何かを水で満たした経験は皆ありますから どれだけかかるか子供に答えさせます 数学も会話もうまくできず、話に入るのを 怖がっている子もいます そこでボードに名前を書き、推測させます すると食いついてくるのです それから今までの過程を踏みます この方法の最もよい点は 教師マニュアル巻末の模範解答から 解答を得るのではないことで その代わりにビデオの結末を見るのですが (笑) これは恐ろしいことですよ マニュアルの模範解答で 常にうまくいく理論モデルは それはそれでいいでしょう でも理論が現実に合わないとき 誤りの原因を話すのは怖いものです それでも授業中の会話は 最も貴重なものです

生徒は楽しんで学習しました クラスの初日にウイルスに 感染した子たちです 今新学期に入っていますが 全く新しいもの見たことのないものを ボードに書くと 年度当初よりも3、4分長く 議論します とても楽しいですよ もう文章題を嫌っていません どんなものか再定義しましたから もう数学を怖がっていません ゆっくりと再定義しているからです これは素晴らしいことです

数学教師はマルチメディアを使って下さい 現実世界を教室に引き入れるからです 高解像度とフルカラーでです 同一条件で直観力を開発させます できる限りの短い質問をさせ 特定の質問が議論で出てくるようにし 生徒自身に問題を作らせます アインシュタインもそう主張しています 最終的に生徒の手助けはしません 教科書の方法は間違っています 問題解決と数的推論を教える義務感から 自らを解き放ち生徒の手助けをしません

今数学教師であるのが素晴らしい理由は 上質の教科課程のツールを 手にしているからです 至るところにあり、安価です オープンライセンスで 自由配布されるこの道具が 現在こんなに安く簡単に手に入るのです ブログにビデオシリーズを載せたところ 2週間で6,000件のビューがありました ある国の教師はメールでこう言います 「素晴らしい。実のある議論になりました ところで、こう改良したらどうでしょう」 素晴らしい提案で 最近ブログにこの問題を載せました 店でどのレジの列に入るかです カート1台に19品の者と 4台のカートに3、5、2、1品の者です この線形モデルは教室でも利用でき 「グッドモーニング アメリカ」に出演しました 変ですよね

ここから結論できるのは 生徒だけではなく、人はー こういうことを切望しているのです 数学は世の中を理解します 数学は直観力の 表現手段です 教育にどんな利害関係があろうとも 生徒 保護者 教師 政治家であろうとも よりよい教科課程を作成してほしいのです 忍耐強く問題解決する人が必要なのです

― もっと見る ―
― 折りたたむ ―


  • 主語
  • 動詞
  • 助動詞
  • 準動詞
  • 関係詞等