Math class needs a makeover | Dan Meyer

ダン・メイヤー：「数学クラス改造計画」

翻訳: Hideki Kamiya 校正: Yuko Osugi

Can I ask you to please recall a time

大好きなものを

when you really loved something --

思い出して下さい

a movie, an album, a song or a book --

映画 アルバム 歌 本など

and you recommended it wholeheartedly

それを好きな人に

to someone you also really liked,

本気で勧めたとします

and you anticipated that reaction, you waited for it,

相手の反応を期待して待ちますが

and it came back, and the person hated it?

返事は正反対、という結末

So, by way of introduction,

前置きなのですが

that is the exact same state

この状態こそまさしく

in which I spent every working day of the last six years. (Laughter)

6年間毎日経験した事です

I teach high school math.

私は高校数学教師です

I sell a product to a market

要らない人に

that doesn't want it, but is forced by law to buy it.

法律なので、とモノを売るのです

I mean, it's just a losing proposition.

失敗元々の商売です

So there's a useful stereotype about students that I see,

説明に便利な生徒の典型例があります

a useful stereotype about you all.

皆さんにも当てはまるかもしれません

I could give you guys

代数IIの

an algebra-two final exam,

期末試験を行うとします

and I would expect no higher

合格率は

than a 25 percent pass rate.

25％以下だと思います

And both of these facts say less about you or my students

これは皆さんや生徒の事より

than they do about what we call math education

アメリカの数学教育の現状を

in the U.S. today.

物語っています

To start with, I'd like to break math down into two categories.

まず数学を２つに分類します

One is computation; this is the stuff you've forgotten.

1つは計算。お忘れだと思います

For example, factoring quadratics with

例えば、2次方程式の因数分解です

leading coefficients greater than one.

1より大きい首位係数のものです

This stuff is also really easy to relearn,

この手の問題は

provided you have a really strong grounding

数的推論の基本が出来ていれば

in reasoning. Math reasoning --

すぐ思い出せます

we'll call it the application

世界に対する

of math processes to the world around us --

数学的プロセスの適用です

this is hard to teach.

教えるのは難しいですが

This is what we would love students to retain,

数学を志すかどうかに関わらず

even if they don't go into mathematical fields.

生徒に身につけて欲しい項目です

This is also something that, the way we teach it in the U.S.

現在のアメリカの教え方では

all but ensures they won't retain it.

まず身に付かないでしょう

So, I'd like to talk about why that is,

今日のテーマは

why that's such a calamity for society, what we can do about it

そんな教育に対して何ができるか、です

and, to close with, why this is an amazing time

なぜ現代が数学教師にとって

to be a math teacher.

驚くべき時代なのかにも触れます

So first, five symptoms

学校で間違った方法で

that you're doing math reasoning wrong

数的推理が教えられて兆候が

in your classroom.

5つあります

One is a lack of initiative; your students don't self-start.

1つは率先力欠如で、自ら学習しません

You finish your lecture block

説明が終わるとすぐに

and immediately you have five hands going up

生徒5人が手を挙げ

asking you to re-explain the entire thing at their desks.

また全て説明してくれと言うのです

Students lack perseverance.

忍耐力も欠けています

They lack retention; you find yourself

記憶力もありません

re-explaining concepts three months later, wholesale.

3ヵ月後にはまた全部説明です

There's an aversion to word problems,

文章題への嫌悪感もあります

which describes 99 percent of my students.

生徒の99％はそうです

And then the other one percent

残りの1％は

is eagerly looking for the formula

公式が当てはまらないかと

to apply in that situation.

一生懸命に探っています

This is really destructive.

目も当てられない状況です

David Milch, creator of "Deadwood" and other amazing TV shows,

「デッドウッド」の創作者のデイビッド ミルチ氏は

has a really good description for this.

これを上手く説明します

He swore off creating

彼は

contemporary drama,

現代を舞台にした

shows set in the present day,

現代劇の創作を止めました

because he saw that when people fill their mind

「チャーリーシーンのハーパーボーイズ」を

with four hours a day of, for example, "Two and a Half Men," no disrespect,

毎日4時間も見ていると

it shapes the neural pathways, he said,

特殊な神経経路ができ

in such a way that they expect simple problems.

単純問題を期待するようになるというのです

He called it, "an impatience with irresolution."

彼の言う「決断できないじれったさ」です

You're impatient with things that don't resolve quickly.

すぐ解決しないものに我慢できないのです

You expect sitcom-sized problems that wrap up in 22 minutes,

22分で問題解決するホームコメディー式を望みます

three commercial breaks and a laugh track.

CM3回と笑い声付きです

And I'll put it to all of you,

このように話していますが

what you already know, that no problem worth solving is that simple.

解決価値のある問題は単純ではありません

I am very concerned about this

心配しているのは

because I'm going to retire in a world that my students will run.

将来生徒の指揮下で暮らすのですよ

I'm doing bad things

自分自身の未来と福祉に

to my own future and well-being

役に立たないことをしているのです

when I teach this way.

教えながら

I'm here to tell you that the way our textbooks -- particularly

現在の教科書、特にー

mass-adopted textbooks -- teach math reasoning

全国版の教科書で数的推論と

and patient problem solving,

問題解決を教えるなら

it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.

「ハーパーボーイズ」を観るのと同じです

(Laughter)

（笑）

In all seriousness. Here's an example from a physics textbook.

さてこれが例です。物理の教科書です

It applies equally to math.

数学に当てはまります

Notice, first of all here,

まず見て頂きたいのは

that you have exactly three pieces of information there,

この3つの情報で

each of which will figure into a formula

この1つ1つがやがて公式となり

somewhere, eventually,

最終的には生徒が

which the student will then compute.

それを使って計算します

I believe in real life.

現実生活では違います

And ask yourself, what problem have you solved, ever,

解決価値のある問題は

that was worth solving

どんなものを解決したことがありますか

where you knew all of the given information in advance;

情報を全て入手している場合ですよ

where you didn't have a surplus of information and you had to filter it out,

情報が余分にあれば取捨選択したでしょう

or you didn't have sufficient information

情報が不十分なら

and had to go find some.

自ら探しに行ったはずです

I'm sure we all agree that no problem worth solving is like that.

有価値の問題でそのようなものは皆無です

And the textbook, I think, knows how it's hamstringing students

教科書も生徒の思考力を阻害しています

because, watch this, this is the practice problem set.

理由はこれをご覧下さい。練習問題です

When it comes time to do the actual problem set,

問題に取り組む段になると

we have problems like this right here

こういう問題が控えています

where we're just swapping out numbers and tweaking the context a little bit.

数字と内容を少しいじっただけです

And if the student still doesn't recognize the stamp this was molded from,

それでも出題元がわからない場合は

it helpfully explains to you

公式が載っている例題を

what sample problem you can return to to find the formula.

親切に教えてくれます

You could literally, I mean this,

そうやって１つの単元を

pass this particular unit without knowing any physics,

文字通り通過します。物理は全く学ばず

just knowing how to decode a textbook. That's a shame.

教科書のどこに何があるかだけを学びます

So I can diagnose the problem a little more specifically in math.

もう少し具体的に数学で検討しましょう

Here's a really cool problem. I like this.

ここに好都合な問題があります

It's about defining steepness and slope

スキーリフトを使って

using a ski lift.

傾斜角とスロープの長さを求めます

But what you have here is actually four separate layers,

4段階の手順があります

and I'm curious which of you can see the four separate layers

どなたかその4つの手順がわかりますか

and, particularly, how when they're compressed together

生徒には凝縮して

and presented to the student all at once,

一度に提示されるのですよ

how that creates this impatient problem solving.

それでじれったい問題解決が起こります

I'll define them here: You have the visual.

実際に解いてみましょう。図があります

You also have the mathematical structure,

皆さんは数学の基礎知識もお持ちです

talking about grids, measurements, labels,

格子、長さ、名称

points, axes, that sort of thing.

点、軸などのことです

You have substeps, which all lead to what we really want to talk about:

また分割して考える手順もご存知です

which section is the steepest.

どの部分が最も険しいですか

So I hope you can see.

話が見えてきたと思います

I really hope you can see how what we're doing here

私達は誘導的な問いと

is taking a compelling question, a compelling answer,

解答という形をとっています

but we're paving a smooth, straight path

しかし問いから問いへと

from one to the other

まっすぐな道を整備してやり

and congratulating our students for how well

途中の小さな割れ目をうまく

they can step over the small cracks in the way.

踏み越えれば褒めてやります

That's all we're doing here.

ただそういうことなのです

So I want to put to you that if we can separate these in a different way

もし違うやり方でこれらを分解して

and build them up with students,

生徒と一緒に組み立てるなら

we can have everything we're looking for in terms of patient problem solving.

まさに問題解決の理想形になるのです

So right here I start with the visual,

この図から始めましょう

and I immediately ask the question:

すぐに質問をします

Which section is the steepest?

どの部分が最も険しいですか

And this starts conversation

こう切り出すのは

because the visual is created in such a way where you can defend two answers.

解答が2つあるようになっているからです

So you get people arguing against each other,

こうして互いに議論させます

friend versus friend,

友人同士で

in pairs, journaling, whatever.

ペアになって記録もとります

And then eventually we realize

そのうち気づきます

it's getting annoying to talk about

スキーヤーは目障りなだけだと

the skier in the lower left-hand side of the screen

画面の左下あるいは

or the skier just above the mid line.

真ん中より少し上の絵です

And we realize how great would it be

またこう感じます

if we just had some A, B, C and D labels

Ａ～Ｄに名称があれば

to talk about them more easily.

話が簡単になるのにと

And then as we start to define what does steepness mean,

勾配の意味を定義し始めるとき

we realize it would be nice to have some measurements

その基準となる数値が

to really narrow it down, specifically what that means.

あればいいと気づきます

And then and only then,

そのとき初めて

we throw down that mathematical structure.

数学的思考法を投げ与えます

The math serves the conversation,

数学は議論に役立ちますが

the conversation doesn't serve the math.

議論は数学に役立ちません

And at that point, I'll put it to you that nine out of 10 classes

その時点で10クラスの内9クラスが

are good to go on the whole slope, steepness thing.

問題への準備ができています

But if you need to,

必要なら生徒自身が

your students can then develop those substeps together.

小ステップを考えることもできます

Do you guys see how this, right here, compared to that --

これが辛抱強い問題解決

which one creates that patient problem solving, that math reasoning?

数的推論を生み出すのです

It's been obvious in my practice, to me.

私の実践では明白でした

And I'll yield the floor here for a second to Einstein,

ここでアインシュタイン君に発言権です

who, I believe, has paid his dues.

これは当然ですよね

He talked about the formulation of a problem being so incredibly important,

彼は問題の定式化が肝要だと言います

and yet in my practice, in the U.S. here,

でも米国の私の授業では

we just give problems to students;

ただ生徒に問題を与えるだけです

we don't involve them in the formulation of the problem.

生徒は問題の定式化に関わりません

So 90 percent of what I do

週5時間の準備時間の

with my five hours of prep time per week

90％を使い

is to take fairly compelling elements

かなり誘導的要素をもつ問題を

of problems like this from my textbook

教科書から選んで

and rebuild them in a way that supports math reasoning and patient problem solving.

数的推論と問題解決の練習用に再編します

And here's how it works.

それはこういうことです

I like this question. It's about a water tank.

水タンクの良問です

The question is: How long will it take you to fill it up?

満たすのにどれだけ時間がかかりますか

First things first, we eliminate all the substeps.

真っ先に小ステップを排除します

Students have to develop those,

生徒が自分で考え

they have to formulate those.

定式化しなければなりません

And then notice that all the information written on there is stuff you'll need.

そこにある全情報が必要と気づくでしょう

None of it's a distractor, so we lose that.

どれも解答の邪魔になりません

Students need to decide, "All right, well,

ここで生徒が考えますー

does the height matter? Does the side of it matter?

高さや大きさは大事かな？

Does the color of the valve matter? What matters here?"

バルブの色は？ここで何が重要でしょう

Such an underrepresented question in math curriculum.

教科課程では提示されていません

So now we have a water tank.

ここに水タンクがあります

How long will it take you to fill it up? And that's it.

満たすのに時間は？それだけです

And because this is the 21st century

21世紀である現在

and we would love to talk about the real world on its own terms,

私達は現実世界をありのままに見ます

not in terms of line art or clip art

それは教科書でよく目にする線アートや

that you so often see in textbooks,

クリップアートではありません

we go out and we take a picture of it.

外に出て、写真を撮ります

So now we have the real deal.

現実を扱っているのです

How long will it take it to fill it up?

満たすのにどれだけかかりますか

And then even better is we take a video,

ビデオを撮ればもっといいでしょう

a video of someone filling it up.

誰かがタンクを満たしているところです

And it's filling up slowly, agonizingly slowly.

腹立たしいほどゆっくりで

It's tedious.

退屈です

Students are looking at their watches, rolling their eyes,

生徒は時計を見ています

and they're all wondering at some point or another,

ある時点でこう思います

"Man, how long is it going to take to fill up?"

「おい、一体どれだけかかるんだよ」

(Laughter)

（笑）

That's how you know you've baited the hook, right?

こうして問題に食いついたのです

And that question, off this right here, is really fun for me

ここからが本当に面白いのです

because, like the intro,

冒頭で言いましたように

I teach kids -- because of my inexperience --

私は経験が浅いので

I teach the kids that are the most remedial, all right?

最も遅れている子供に教えています

And I've got kids who will not join a conversation about math

数学の話には参加しない子供もいます

because someone else has the formula;

誰かが公式を知っており

someone else knows how to work the formula better than me,

私よりその使い方を知っているからです

so I won't talk about it.

だからそれについて話しません

But here, every student is on a level playing field of intuition.

でもここでは皆同じ直観力を持っています

Everyone's filled something up with water before,

何かを水で満たした経験は皆ありますから

so I get kids answering the question, "How long will it take?"

どれだけかかるか子供に答えさせます

I've got kids who are mathematically and conversationally intimidated

数学も会話もうまくできず、話に入るのを

joining the conversation.

怖がっている子もいます

We put names on the board, attach them to guesses,

そこでボードに名前を書き、推測させます

and kids have bought in here.

すると食いついてくるのです

And then we follow the process I've described.

それから今までの過程を踏みます

And the best part here, or one of the better parts

この方法の最もよい点は

is that we don't get our answer from the answer key

教師マニュアル巻末の模範解答から

in the back of the teacher's edition.

解答を得るのではないことで

We, instead, just watch the end of the movie.

その代わりにビデオの結末を見るのですが

(Laughter)

（笑）

And that's terrifying,

これは恐ろしいことですよ

because the theoretical models that always work out

マニュアルの模範解答で

in the answer key in the back of a teacher's edition,

常にうまくいく理論モデルは

that's great, but

それはそれでいいでしょう

it's scary to talk about sources of error

でも理論が現実に合わないとき

when the theoretical does not match up with the practical.

誤りの原因を話すのは怖いものです

But those conversations have been so valuable,

それでも授業中の会話は

among the most valuable.

最も貴重なものです

So I'm here to report some really fun games

生徒は楽しんで学習しました

with students who come pre-installed

クラスの初日にウイルスに

with these viruses day one of the class.

感染した子たちです

These are the kids who now, one semester in,

今新学期に入っていますが

I can put something on the board,

全く新しいもの見たことのないものを

totally new, totally foreign,

ボードに書くと

and they'll have a conversation about it for three or four minutes more

年度当初よりも3、4分長く

than they would have at the start of the year,

議論します

which is just so fun.

とても楽しいですよ

We're no longer averse to word problems,

もう文章題を嫌っていません

because we've redefined what a word problem is.

どんなものか再定義しましたから

We're no longer intimidated by math,

もう数学を怖がっていません

because we're slowly redefining what math is.

ゆっくりと再定義しているからです

This has been a lot of fun.

これは素晴らしいことです

I encourage math teachers I talk to to use multimedia,

数学教師はマルチメディアを使って下さい

because it brings the real world into your classroom

現実世界を教室に引き入れるからです

in high resolution and full color;

高解像度とフルカラーでです

to encourage student intuition for that level playing field;

同一条件で直観力を開発させます

to ask the shortest question you possibly can

できる限りの短い質問をさせ

and let those more specific questions come out in conversation;

特定の質問が議論で出てくるようにし

to let students build the problem,

生徒自身に問題を作らせます

because Einstein said so;

アインシュタインもそう主張しています

and to finally, in total, just be less helpful,

最終的に生徒の手助けはしません

because the textbook is helping you in all the wrong ways:

教科書の方法は間違っています

It's buying you out of your obligation,

問題解決と数的推論を教える義務感から

for patient problem solving and math reasoning, to be less helpful.

自らを解き放ち生徒の手助けをしません

And why this is an amazing time to be a math teacher right now

今数学教師であるのが素晴らしい理由は

is because we have the tools to create

上質の教科課程のツールを

this high-quality curriculum in our front pocket.

手にしているからです

It's ubiquitous and fairly cheap,

至るところにあり、安価です

and the tools to distribute it

オープンライセンスで

freely under open licenses

自由配布されるこの道具が

has also never been cheaper or more ubiquitous.

現在こんなに安く簡単に手に入るのです

I put a video series on my blog not so long ago

ブログにビデオシリーズを載せたところ

and it got 6,000 views in two weeks.

2週間で6,000件のビューがありました

I get emails still from teachers in countries I've never visited

ある国の教師はメールでこう言います

saying, "Wow, yeah. We had a good conversation about that.

「素晴らしい。実のある議論になりました

Oh, and by the way, here's how I made your stuff better,"

ところで、こう改良したらどうでしょう」

which, wow.

素晴らしい提案で

I put this problem on my blog recently:

最近ブログにこの問題を載せました

In a grocery store, which line do you get into,

店でどのレジの列に入るかです

the one that has one cart and 19 items

カート1台に19品の者と

or the line with four carts and three, five, two and one items.

4台のカートに3、5、2、1品の者です

And the linear modeling involved in that was some good stuff for my classroom,

この線形モデルは教室でも利用でき

but it eventually got me on "Good Morning America" a few weeks later,

「グッドモーニング アメリカ」に出演しました

which is just bizarre, right?

変ですよね

And from all of this, I can only conclude

ここから結論できるのは

that people, not just students,

生徒だけではなく、人はー

are really hungry for this.

こういうことを切望しているのです

Math makes sense of the world.

数学は世の中を理解します

Math is the vocabulary

数学は直観力の

for your own intuition.

表現手段です

So I just really encourage you, whatever your stake is in education --

教育にどんな利害関係があろうとも

whether you're a student, parent, teacher, policy maker, whatever --

生徒 保護者 教師 政治家であろうとも

insist on better math curriculum.

よりよい教科課程を作成してほしいのです

We need more patient problem solvers. Thank you. (Applause)

忍耐強く問題解決する人が必要なのです