when you really loved something --

大好きなものを

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

思い出して下さい

and you recommended it wholeheartedly

映画 アルバム 歌 本など

to someone you also really liked,

それを好きな人に

and you anticipated that reaction, you waited for it,

本気で勧めたとします

and it came back, and the person hated it?

相手の反応を期待して待ちますが

So, by way of introduction,

返事は正反対、という結末

that is the exact same state

前置きなのですが

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

この状態こそまさしく

I teach high school math.

6年間毎日経験した事です

I sell a product to a market

私は高校数学教師です

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

要らない人に

I mean, it's just a losing proposition.

法律なので、とモノを売るのです

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

失敗元々の商売です

a useful stereotype about you all.

説明に便利な生徒の典型例があります

I could give you guys

皆さんにも当てはまるかもしれません

an algebra-two final exam,

代数IIの

and I would expect no higher

期末試験を行うとします

than a 25 percent pass rate.

合格率は

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

25％以下だと思います

than they do about what we call math education

これは皆さんや生徒の事より

in the U.S. today.

アメリカの数学教育の現状を

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

物語っています

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

まず数学を２つに分類します

For example, factoring quadratics with

1つは計算。お忘れだと思います

leading coefficients greater than one.

例えば、2次方程式の因数分解です

This stuff is also really easy to relearn,

1より大きい首位係数のものです

provided you have a really strong grounding

この手の問題は

in reasoning. Math reasoning --

数的推論の基本が出来ていれば

we'll call it the application

すぐ思い出せます

of math processes to the world around us --

世界に対する

this is hard to teach.

数学的プロセスの適用です

This is what we would love students to retain,

教えるのは難しいですが

even if they don't go into mathematical fields.

数学を志すかどうかに関わらず

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

生徒に身につけて欲しい項目です

all but ensures they won't retain it.

現在のアメリカの教え方では

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

まず身に付かないでしょう

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

今日のテーマは

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

そんな教育に対して何ができるか、です

to be a math teacher.

なぜ現代が数学教師にとって

So first, five symptoms

驚くべき時代なのかにも触れます

that you're doing math reasoning wrong

学校で間違った方法で

in your classroom.

数的推理が教えられて兆候が

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

5つあります

You finish your lecture block

1つは率先力欠如で、自ら学習しません

and immediately you have five hands going up

説明が終わるとすぐに

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

生徒5人が手を挙げ

Students lack perseverance.

また全て説明してくれと言うのです

They lack retention; you find yourself

忍耐力も欠けています

re-explaining concepts three months later, wholesale.

記憶力もありません

There's an aversion to word problems,

3ヵ月後にはまた全部説明です

which describes 99 percent of my students.

文章題への嫌悪感もあります

And then the other one percent

生徒の99％はそうです

is eagerly looking for the formula

残りの1％は

to apply in that situation.

公式が当てはまらないかと

This is really destructive.

一生懸命に探っています

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

目も当てられない状況です

has a really good description for this.

「デッドウッド」の創作者のデイビッド ミルチ氏は

He swore off creating

これを上手く説明します

contemporary drama,

彼は

shows set in the present day,

現代を舞台にした

because he saw that when people fill their mind

現代劇の創作を止めました

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

「チャーリーシーンのハーパーボーイズ」を

it shapes the neural pathways, he said,

毎日4時間も見ていると

in such a way that they expect simple problems.

特殊な神経経路ができ

He called it, "an impatience with irresolution."

単純問題を期待するようになるというのです

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

彼の言う「決断できないじれったさ」です

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

すぐ解決しないものに我慢できないのです

three commercial breaks and a laugh track.

22分で問題解決するホームコメディー式を望みます

And I'll put it to all of you,

CM3回と笑い声付きです

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

このように話していますが

I am very concerned about this

解決価値のある問題は単純ではありません

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

心配しているのは

I'm doing bad things

将来生徒の指揮下で暮らすのですよ

to my own future and well-being

自分自身の未来と福祉に

when I teach this way.

役に立たないことをしているのです

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

教えながら

mass-adopted textbooks -- teach math reasoning

現在の教科書、特にー

and patient problem solving,

全国版の教科書で数的推論と

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

問題解決を教えるなら

(Laughter)

「ハーパーボーイズ」を観るのと同じです

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

（笑）

It applies equally to math.

さてこれが例です。物理の教科書です

Notice, first of all here,

数学に当てはまります

that you have exactly three pieces of information there,

まず見て頂きたいのは

each of which will figure into a formula

この3つの情報で

somewhere, eventually,

この1つ1つがやがて公式となり

which the student will then compute.

最終的には生徒が

I believe in real life.

それを使って計算します

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

現実生活では違います

that was worth solving

解決価値のある問題は

where you knew all of the given information in advance;

どんなものを解決したことがありますか

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

情報を全て入手している場合ですよ

or you didn't have sufficient information

情報が余分にあれば取捨選択したでしょう

and had to go find some.

情報が不十分なら

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

自ら探しに行ったはずです

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

有価値の問題でそのようなものは皆無です

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

教科書も生徒の思考力を阻害しています

When it comes time to do the actual problem set,

理由はこれをご覧下さい。練習問題です

we have problems like this right here

問題に取り組む段になると

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

こういう問題が控えています

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

数字と内容を少しいじっただけです

it helpfully explains to you

それでも出題元がわからない場合は

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

公式が載っている例題を

You could literally, I mean this,

親切に教えてくれます

pass this particular unit without knowing any physics,

そうやって１つの単元を

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

文字通り通過します。物理は全く学ばず

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

教科書のどこに何があるかだけを学びます

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

もう少し具体的に数学で検討しましょう

It's about defining steepness and slope

ここに好都合な問題があります

using a ski lift.

スキーリフトを使って

But what you have here is actually four separate layers,

傾斜角とスロープの長さを求めます

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

4段階の手順があります

and, particularly, how when they're compressed together

どなたかその4つの手順がわかりますか

and presented to the student all at once,

生徒には凝縮して

how that creates this impatient problem solving.

一度に提示されるのですよ

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

それでじれったい問題解決が起こります

You also have the mathematical structure,

実際に解いてみましょう。図があります

talking about grids, measurements, labels,

皆さんは数学の基礎知識もお持ちです

points, axes, that sort of thing.

格子、長さ、名称

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

点、軸などのことです

which section is the steepest.

また分割して考える手順もご存知です

So I hope you can see.

どの部分が最も険しいですか

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

話が見えてきたと思います

is taking a compelling question, a compelling answer,

私達は誘導的な問いと

but we're paving a smooth, straight path

解答という形をとっています

from one to the other

しかし問いから問いへと

and congratulating our students for how well

まっすぐな道を整備してやり

they can step over the small cracks in the way.

途中の小さな割れ目をうまく

That's all we're doing here.

踏み越えれば褒めてやります

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

ただそういうことなのです

and build them up with students,

もし違うやり方でこれらを分解して

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

生徒と一緒に組み立てるなら

So right here I start with the visual,

まさに問題解決の理想形になるのです

and I immediately ask the question:

この図から始めましょう

Which section is the steepest?

すぐに質問をします

And this starts conversation

どの部分が最も険しいですか

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

こう切り出すのは

So you get people arguing against each other,

解答が2つあるようになっているからです

friend versus friend,

こうして互いに議論させます

in pairs, journaling, whatever.

友人同士で

And then eventually we realize

ペアになって記録もとります

it's getting annoying to talk about

そのうち気づきます

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

スキーヤーは目障りなだけだと

or the skier just above the mid line.

画面の左下あるいは

And we realize how great would it be

真ん中より少し上の絵です

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

またこう感じます

to talk about them more easily.

Ａ～Ｄに名称があれば

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

話が簡単になるのにと

we realize it would be nice to have some measurements

勾配の意味を定義し始めるとき

to really narrow it down, specifically what that means.

その基準となる数値が

And then and only then,

あればいいと気づきます

we throw down that mathematical structure.

そのとき初めて

The math serves the conversation,

数学的思考法を投げ与えます

the conversation doesn't serve the math.

数学は議論に役立ちますが

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

議論は数学に役立ちません

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

その時点で10クラスの内9クラスが

But if you need to,

問題への準備ができています

your students can then develop those substeps together.

必要なら生徒自身が

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

小ステップを考えることもできます

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

これが辛抱強い問題解決

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

数的推論を生み出すのです

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

私の実践では明白でした

who, I believe, has paid his dues.

ここでアインシュタイン君に発言権です

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

これは当然ですよね

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

彼は問題の定式化が肝要だと言います

we just give problems to students;

でも米国の私の授業では

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

ただ生徒に問題を与えるだけです

So 90 percent of what I do

生徒は問題の定式化に関わりません

with my five hours of prep time per week

週5時間の準備時間の

is to take fairly compelling elements

90％を使い

of problems like this from my textbook

かなり誘導的要素をもつ問題を

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

教科書から選んで

And here's how it works.

数的推論と問題解決の練習用に再編します

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

それはこういうことです

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

水タンクの良問です

First things first, we eliminate all the substeps.

満たすのにどれだけ時間がかかりますか

Students have to develop those,

真っ先に小ステップを排除します

they have to formulate those.

生徒が自分で考え

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

定式化しなければなりません

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

そこにある全情報が必要と気づくでしょう

Students need to decide, "All right, well,

どれも解答の邪魔になりません

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

ここで生徒が考えますー

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

高さや大きさは大事かな？

Such an underrepresented question in math curriculum.

バルブの色は？ここで何が重要でしょう

So now we have a water tank.

教科課程では提示されていません

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

ここに水タンクがあります

And because this is the 21st century

満たすのに時間は？それだけです

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

21世紀である現在

not in terms of line art or clip art

私達は現実世界をありのままに見ます

that you so often see in textbooks,

それは教科書でよく目にする線アートや

we go out and we take a picture of it.

クリップアートではありません

So now we have the real deal.

外に出て、写真を撮ります

How long will it take it to fill it up?

現実を扱っているのです

And then even better is we take a video,

満たすのにどれだけかかりますか

a video of someone filling it up.

ビデオを撮ればもっといいでしょう

And it's filling up slowly, agonizingly slowly.

誰かがタンクを満たしているところです

It's tedious.

腹立たしいほどゆっくりで

Students are looking at their watches, rolling their eyes,

退屈です

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

生徒は時計を見ています

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

ある時点でこう思います

(Laughter)

「おい、一体どれだけかかるんだよ」

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

（笑）

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

こうして問題に食いついたのです

because, like the intro,

ここからが本当に面白いのです

I teach kids -- because of my inexperience --

冒頭で言いましたように

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

私は経験が浅いので

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

最も遅れている子供に教えています

because someone else has the formula;

数学の話には参加しない子供もいます

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

誰かが公式を知っており

so I won't talk about it.

私よりその使い方を知っているからです

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

だからそれについて話しません

Everyone's filled something up with water before,

でもここでは皆同じ直観力を持っています

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

何かを水で満たした経験は皆ありますから

I've got kids who are mathematically and conversationally intimidated

どれだけかかるか子供に答えさせます

joining the conversation.

数学も会話もうまくできず、話に入るのを

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

怖がっている子もいます

and kids have bought in here.

そこでボードに名前を書き、推測させます

And then we follow the process I've described.

すると食いついてくるのです

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

それから今までの過程を踏みます

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

この方法の最もよい点は

in the back of the teacher's edition.

教師マニュアル巻末の模範解答から

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

解答を得るのではないことで

(Laughter)

その代わりにビデオの結末を見るのですが

And that's terrifying,

（笑）

because the theoretical models that always work out

これは恐ろしいことですよ

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

マニュアルの模範解答で

that's great, but

常にうまくいく理論モデルは

it's scary to talk about sources of error

それはそれでいいでしょう

when the theoretical does not match up with the practical.

でも理論が現実に合わないとき

But those conversations have been so valuable,

誤りの原因を話すのは怖いものです

among the most valuable.

それでも授業中の会話は

So I'm here to report some really fun games

最も貴重なものです

with students who come pre-installed