TEDxNYED
Dan Meyer: Math class needs a makeover
ダン・メイヤー：「数学クラス改造計画」
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今日の数学はペイント・バイ・ナンバー型演習が当然で、その演習が得意になるように教えられており、子供は問題を解くよりも大切な定式化能力を失っている。TED×NYEDでダン・メイヤーは立ち止まって考える実証済み数学教授法を披露してくれる。
Double-click the English transcript below to play the video.
00:15
Can I ask you to please recall a time
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大好きなものを
00:18
when you really loved something --
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思い出して下さい
00:20
a movie, an album, a song or a book --
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映画 アルバム 歌 本など
00:22
and you recommended it wholeheartedly
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それを好きな人に
00:25
to someone you also really liked,
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本気で勧めたとします
00:27
and you anticipated that reaction, you waited for it,
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相手の反応を期待して待ちますが
00:29
and it came back, and the person hated it?
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返事は正反対、という結末
00:32
So, by way of introduction,
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前置きなのですが
00:34
that is the exact same state
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この状態こそまさしく
00:36
in which I spent every working day of the last six years. (Laughter)
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6年間毎日経験した事です
00:39
I teach high school math.
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私は高校数学教師です
00:41
I sell a product to a market
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要らない人に
00:44
that doesn't want it, but is forced by law to buy it.
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法律なので、とモノを売るのです
00:47
I mean, it's just a losing proposition.
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失敗元々の商売です
00:50
So there's a useful stereotype about students that I see,
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説明に便利な生徒の典型例があります
00:53
a useful stereotype about you all.
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皆さんにも当てはまるかもしれません
00:55
I could give you guys
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代数IIの
00:57
an algebra-two final exam,
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期末試験を行うとします
00:59
and I would expect no higher
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合格率は
01:01
than a 25 percent pass rate.
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25％以下だと思います
01:03
And both of these facts say less about you or my students
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これは皆さんや生徒の事より
01:06
than they do about what we call math education
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アメリカの数学教育の現状を
01:08
in the U.S. today.
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物語っています
01:10
To start with, I'd like to break math down into two categories.
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まず数学を２つに分類します
01:13
One is computation; this is the stuff you've forgotten.
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1つは計算。お忘れだと思います
01:16
For example, factoring quadratics with
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例えば、2次方程式の因数分解です
01:18
leading coefficients greater than one.
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1より大きい首位係数のものです
01:20
This stuff is also really easy to relearn,
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この手の問題は
01:22
provided you have a really strong grounding
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数的推論の基本が出来ていれば
01:24
in reasoning. Math reasoning --
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すぐ思い出せます
01:26
we'll call it the application
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世界に対する
01:28
of math processes to the world around us --
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数学的プロセスの適用です
01:30
this is hard to teach.
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教えるのは難しいですが
01:32
This is what we would love students to retain,
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数学を志すかどうかに関わらず
01:34
even if they don't go into mathematical fields.
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生徒に身につけて欲しい項目です
01:36
This is also something that, the way we teach it in the U.S.
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現在のアメリカの教え方では
01:38
all but ensures they won't retain it.
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まず身に付かないでしょう
01:41
So, I'd like to talk about why that is,
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今日のテーマは
01:42
why that's such a calamity for society, what we can do about it
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そんな教育に対して何ができるか、です
01:45
and, to close with, why this is an amazing time
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なぜ現代が数学教師にとって
01:47
to be a math teacher.
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驚くべき時代なのかにも触れます
01:49
So first, five symptoms
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学校で間違った方法で
01:51
that you're doing math reasoning wrong
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数的推理が教えられて兆候が
01:53
in your classroom.
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5つあります
01:55
One is a lack of initiative; your students don't self-start.
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1つは率先力欠如で、自ら学習しません
01:58
You finish your lecture block
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説明が終わるとすぐに
02:00
and immediately you have five hands going up
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生徒5人が手を挙げ
02:02
asking you to re-explain the entire thing at their desks.
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また全て説明してくれと言うのです
02:04
Students lack perseverance.
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忍耐力も欠けています
02:06
They lack retention; you find yourself
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記憶力もありません
02:08
re-explaining concepts three months later, wholesale.
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3ヵ月後にはまた全部説明です
02:10
There's an aversion to word problems,
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文章題への嫌悪感もあります
02:12
which describes 99 percent of my students.
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生徒の99％はそうです
02:14
And then the other one percent
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残りの1％は
02:16
is eagerly looking for the formula
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公式が当てはまらないかと
02:18
to apply in that situation.
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一生懸命に探っています
02:20
This is really destructive.
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目も当てられない状況です
02:22
David Milch, creator of "Deadwood" and other amazing TV shows,
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「デッドウッド」の創作者のデイビッド ミルチ氏は
02:25
has a really good description for this.
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これを上手く説明します
02:28
He swore off creating
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彼は
02:30
contemporary drama,
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現代を舞台にした
02:32
shows set in the present day,
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現代劇の創作を止めました
02:34
because he saw that when people fill their mind
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「チャーリーシーンのハーパーボーイズ」を
02:36
with four hours a day of, for example, "Two and a Half Men," no disrespect,
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毎日4時間も見ていると
02:39
it shapes the neural pathways, he said,
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特殊な神経経路ができ
02:41
in such a way that they expect simple problems.
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単純問題を期待するようになるというのです
02:44
He called it, "an impatience with irresolution."
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彼の言う「決断できないじれったさ」です
02:47
You're impatient with things that don't resolve quickly.
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すぐ解決しないものに我慢できないのです
02:50
You expect sitcom-sized problems that wrap up in 22 minutes,
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22分で問題解決するホームコメディー式を望みます
02:53
three commercial breaks and a laugh track.
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CM3回と笑い声付きです
02:56
And I'll put it to all of you,
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このように話していますが
02:59
what you already know, that no problem worth solving is that simple.
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解決価値のある問題は単純ではありません
03:02
I am very concerned about this
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心配しているのは
03:04
because I'm going to retire in a world that my students will run.
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将来生徒の指揮下で暮らすのですよ
03:07
I'm doing bad things
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自分自身の未来と福祉に
03:09
to my own future and well-being
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役に立たないことをしているのです
03:11
when I teach this way.
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教えながら
03:13
I'm here to tell you that the way our textbooks -- particularly
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現在の教科書、特にー
03:16
mass-adopted textbooks -- teach math reasoning
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全国版の教科書で数的推論と
03:19
and patient problem solving,
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問題解決を教えるなら
03:21
it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
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「ハーパーボーイズ」を観るのと同じです
03:24
(Laughter)
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（笑）
03:26
In all seriousness. Here's an example from a physics textbook.
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さてこれが例です。物理の教科書です
03:29
It applies equally to math.
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数学に当てはまります
03:31
Notice, first of all here,
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まず見て頂きたいのは
03:33
that you have exactly three pieces of information there,
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この3つの情報で
03:35
each of which will figure into a formula
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この1つ1つがやがて公式となり
03:37
somewhere, eventually,
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最終的には生徒が
03:39
which the student will then compute.
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それを使って計算します
03:41
I believe in real life.
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現実生活では違います
03:43
And ask yourself, what problem have you solved, ever,
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解決価値のある問題は
03:45
that was worth solving
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どんなものを解決したことがありますか
03:47
where you knew all of the given information in advance;
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情報を全て入手している場合ですよ
03:49
where you didn't have a surplus of information and you had to filter it out,
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情報が余分にあれば取捨選択したでしょう
03:52
or you didn't have sufficient information
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情報が不十分なら
03:54
and had to go find some.
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自ら探しに行ったはずです
03:56
I'm sure we all agree that no problem worth solving is like that.
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有価値の問題でそのようなものは皆無です
03:59
And the textbook, I think, knows how it's hamstringing students
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教科書も生徒の思考力を阻害しています
04:02
because, watch this, this is the practice problem set.
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理由はこれをご覧下さい。練習問題です
04:05
When it comes time to do the actual problem set,
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問題に取り組む段になると
04:07
we have problems like this right here
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こういう問題が控えています
04:09
where we're just swapping out numbers and tweaking the context a little bit.
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数字と内容を少しいじっただけです
04:12
And if the student still doesn't recognize the stamp this was molded from,
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それでも出題元がわからない場合は
04:15
it helpfully explains to you
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公式が載っている例題を
04:17
what sample problem you can return to to find the formula.
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親切に教えてくれます
04:20
You could literally, I mean this,
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そうやって１つの単元を
04:22
pass this particular unit without knowing any physics,
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文字通り通過します。物理は全く学ばず
04:25
just knowing how to decode a textbook. That's a shame.
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教科書のどこに何があるかだけを学びます
04:28
So I can diagnose the problem a little more specifically in math.
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もう少し具体的に数学で検討しましょう
04:31
Here's a really cool problem. I like this.
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ここに好都合な問題があります
04:33
It's about defining steepness and slope
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スキーリフトを使って
04:35
using a ski lift.
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傾斜角とスロープの長さを求めます
04:37
But what you have here is actually four separate layers,
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4段階の手順があります
04:39
and I'm curious which of you can see the four separate layers
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どなたかその4つの手順がわかりますか
04:42
and, particularly, how when they're compressed together
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生徒には凝縮して
04:45
and presented to the student all at once,
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一度に提示されるのですよ
04:47
how that creates this impatient problem solving.
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それでじれったい問題解決が起こります
04:50
I'll define them here: You have the visual.
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実際に解いてみましょう。図があります
04:52
You also have the mathematical structure,
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皆さんは数学の基礎知識もお持ちです
04:54
talking about grids, measurements, labels,
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格子、長さ、名称
04:56
points, axes, that sort of thing.
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点、軸などのことです
04:58
You have substeps, which all lead to what we really want to talk about:
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また分割して考える手順もご存知です
05:01
which section is the steepest.
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どの部分が最も険しいですか
05:03
So I hope you can see.
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話が見えてきたと思います
05:05
I really hope you can see how what we're doing here
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私達は誘導的な問いと
05:07
is taking a compelling question, a compelling answer,
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解答という形をとっています
05:09
but we're paving a smooth, straight path
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しかし問いから問いへと
05:11
from one to the other
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まっすぐな道を整備してやり
05:13
and congratulating our students for how well
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途中の小さな割れ目をうまく
05:15
they can step over the small cracks in the way.
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踏み越えれば褒めてやります
05:17
That's all we're doing here.
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ただそういうことなのです
05:19
So I want to put to you that if we can separate these in a different way
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もし違うやり方でこれらを分解して
05:21
and build them up with students,
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生徒と一緒に組み立てるなら
05:23
we can have everything we're looking for in terms of patient problem solving.
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まさに問題解決の理想形になるのです
05:26
So right here I start with the visual,
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この図から始めましょう
05:28
and I immediately ask the question:
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すぐに質問をします
05:30
Which section is the steepest?
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どの部分が最も険しいですか
05:32
And this starts conversation
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こう切り出すのは
05:34
because the visual is created in such a way where you can defend two answers.
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解答が2つあるようになっているからです
05:37
So you get people arguing against each other,
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こうして互いに議論させます
05:39
friend versus friend,
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友人同士で
05:41
in pairs, journaling, whatever.
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ペアになって記録もとります
05:43
And then eventually we realize
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そのうち気づきます
05:45
it's getting annoying to talk about
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スキーヤーは目障りなだけだと
05:47
the skier in the lower left-hand side of the screen
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画面の左下あるいは
05:49
or the skier just above the mid line.
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真ん中より少し上の絵です
05:51
And we realize how great would it be
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またこう感じます
05:53
if we just had some A, B, C and D labels
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Ａ～Ｄに名称があれば
05:55
to talk about them more easily.
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話が簡単になるのにと
05:57
And then as we start to define what does steepness mean,
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勾配の意味を定義し始めるとき
06:00
we realize it would be nice to have some measurements
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その基準となる数値が
06:02
to really narrow it down, specifically what that means.
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あればいいと気づきます
06:05
And then and only then,
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そのとき初めて
06:07
we throw down that mathematical structure.
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数学的思考法を投げ与えます
06:09
The math serves the conversation,
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数学は議論に役立ちますが
06:11
the conversation doesn't serve the math.
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議論は数学に役立ちません
06:13
And at that point, I'll put it to you that nine out of 10 classes
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その時点で10クラスの内9クラスが
06:16
are good to go on the whole slope, steepness thing.
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問題への準備ができています
06:18
But if you need to,
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必要なら生徒自身が
06:20
your students can then develop those substeps together.
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小ステップを考えることもできます
06:22
Do you guys see how this, right here, compared to that --
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これが辛抱強い問題解決
06:25
which one creates that patient problem solving, that math reasoning?
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数的推論を生み出すのです
06:28
It's been obvious in my practice, to me.
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私の実践では明白でした
06:31
And I'll yield the floor here for a second to Einstein,
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ここでアインシュタイン君に発言権です
06:33
who, I believe, has paid his dues.
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これは当然ですよね
06:35
He talked about the formulation of a problem being so incredibly important,
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彼は問題の定式化が肝要だと言います
06:38
and yet in my practice, in the U.S. here,
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でも米国の私の授業では
06:40
we just give problems to students;
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ただ生徒に問題を与えるだけです
06:42
we don't involve them in the formulation of the problem.
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生徒は問題の定式化に関わりません
06:46
So 90 percent of what I do
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週5時間の準備時間の
06:48
with my five hours of prep time per week
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90％を使い
06:50
is to take fairly compelling elements
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かなり誘導的要素をもつ問題を
06:53
of problems like this from my textbook
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教科書から選んで
06:55
and rebuild them in a way that supports math reasoning and patient problem solving.
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数的推論と問題解決の練習用に再編します
06:58
And here's how it works.
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それはこういうことです
07:00
I like this question. It's about a water tank.
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水タンクの良問です
07:02
The question is: How long will it take you to fill it up?
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満たすのにどれだけ時間がかかりますか
07:04
First things first, we eliminate all the substeps.
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真っ先に小ステップを排除します
07:06
Students have to develop those,
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生徒が自分で考え
07:08
they have to formulate those.
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定式化しなければなりません
07:10
And then notice that all the information written on there is stuff you'll need.
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そこにある全情報が必要と気づくでしょう
07:13
None of it's a distractor, so we lose that.
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どれも解答の邪魔になりません
07:15
Students need to decide, "All right, well,
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ここで生徒が考えますー
07:17
does the height matter? Does the side of it matter?
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高さや大きさは大事かな？
07:19
Does the color of the valve matter? What matters here?"
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バルブの色は？ここで何が重要でしょう
07:22
Such an underrepresented question in math curriculum.
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教科課程では提示されていません
07:25
So now we have a water tank.
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ここに水タンクがあります
07:27
How long will it take you to fill it up? And that's it.
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満たすのに時間は？それだけです
07:29
And because this is the 21st century
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21世紀である現在
07:31
and we would love to talk about the real world on its own terms,
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私達は現実世界をありのままに見ます
07:34
not in terms of line art or clip art
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それは教科書でよく目にする線アートや
07:37
that you so often see in textbooks,
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クリップアートではありません
07:39
we go out and we take a picture of it.
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外に出て、写真を撮ります
07:41
So now we have the real deal.
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現実を扱っているのです
07:43
How long will it take it to fill it up?
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満たすのにどれだけかかりますか
07:45
And then even better is we take a video,
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ビデオを撮ればもっといいでしょう
07:47
a video of someone filling it up.
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誰かがタンクを満たしているところです
07:50
And it's filling up slowly, agonizingly slowly.
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腹立たしいほどゆっくりで
07:52
It's tedious.
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退屈です
07:54
Students are looking at their watches, rolling their eyes,
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生徒は時計を見ています
07:56
and they're all wondering at some point or another,
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ある時点でこう思います
07:59
"Man, how long is it going to take to fill up?"
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「おい、一体どれだけかかるんだよ」
08:02
(Laughter)
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（笑）
08:07
That's how you know you've baited the hook, right?
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こうして問題に食いついたのです
08:11
And that question, off this right here, is really fun for me
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ここからが本当に面白いのです
08:14
because, like the intro,
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冒頭で言いましたように
08:16
I teach kids -- because of my inexperience --
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私は経験が浅いので
08:19
I teach the kids that are the most remedial, all right?
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最も遅れている子供に教えています
08:21
And I've got kids who will not join a conversation about math
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数学の話には参加しない子供もいます
08:24
because someone else has the formula;
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誰かが公式を知っており
08:26
someone else knows how to work the formula better than me,
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私よりその使い方を知っているからです
08:29
so I won't talk about it.
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だからそれについて話しません
08:31
But here, every student is on a level playing field of intuition.
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でもここでは皆同じ直観力を持っています
08:34
Everyone's filled something up with water before,
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何かを水で満たした経験は皆ありますから
08:37
so I get kids answering the question, "How long will it take?"
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どれだけかかるか子供に答えさせます
08:40
I've got kids who are mathematically and conversationally intimidated
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数学も会話もうまくできず、話に入るのを
08:43
joining the conversation.
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怖がっている子もいます
08:45
We put names on the board, attach them to guesses,
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そこでボードに名前を書き、推測させます
08:48
and kids have bought in here.
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すると食いついてくるのです
08:50
And then we follow the process I've described.
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それから今までの過程を踏みます
08:52
And the best part here, or one of the better parts
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この方法の最もよい点は
08:54
is that we don't get our answer from the answer key
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教師マニュアル巻末の模範解答から
08:56
in the back of the teacher's edition.
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解答を得るのではないことで
08:58
We, instead, just watch the end of the movie.
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その代わりにビデオの結末を見るのですが
09:01
(Laughter)
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（笑）
09:03
And that's terrifying,
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これは恐ろしいことですよ
09:05
because the theoretical models that always work out
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マニュアルの模範解答で
09:07
in the answer key in the back of a teacher's edition,
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常にうまくいく理論モデルは
09:09
that's great, but
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それはそれでいいでしょう
09:11
it's scary to talk about sources of error
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でも理論が現実に合わないとき
09:13
when the theoretical does not match up with the practical.
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誤りの原因を話すのは怖いものです
09:15
But those conversations have been so valuable,
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それでも授業中の会話は
09:17
among the most valuable.
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最も貴重なものです
09:19
So I'm here to report some really fun games
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生徒は楽しんで学習しました
09:21
with students who come pre-installed
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クラスの初日にウイルスに
09:23
with these viruses day one of the class.
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感染した子たちです
09:25
These are the kids who now, one semester in,
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今新学期に入っていますが
09:28
I can put something on the board,
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全く新しいもの見たことのないものを
09:30
totally new, totally foreign,
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ボードに書くと
09:32
and they'll have a conversation about it for three or four minutes more
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年度当初よりも3、4分長く
09:34
than they would have at the start of the year,
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議論します
09:36
which is just so fun.
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とても楽しいですよ
09:38
We're no longer averse to word problems,
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もう文章題を嫌っていません
09:41
because we've redefined what a word problem is.
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どんなものか再定義しましたから
09:44
We're no longer intimidated by math,
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もう数学を怖がっていません
09:46
because we're slowly redefining what math is.
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ゆっくりと再定義しているからです
09:48
This has been a lot of fun.
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これは素晴らしいことです
09:50
I encourage math teachers I talk to to use multimedia,
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数学教師はマルチメディアを使って下さい
09:53
because it brings the real world into your classroom
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現実世界を教室に引き入れるからです
09:55
in high resolution and full color;
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高解像度とフルカラーでです
09:57
to encourage student intuition for that level playing field;
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同一条件で直観力を開発させます
10:00
to ask the shortest question you possibly can
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できる限りの短い質問をさせ
10:02
and let those more specific questions come out in conversation;
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特定の質問が議論で出てくるようにし
10:05
to let students build the problem,
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生徒自身に問題を作らせます
10:07
because Einstein said so;
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アインシュタインもそう主張しています
10:09
and to finally, in total, just be less helpful,
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最終的に生徒の手助けはしません
10:12
because the textbook is helping you in all the wrong ways:
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教科書の方法は間違っています
10:14
It's buying you out of your obligation,
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問題解決と数的推論を教える義務感から
10:17
for patient problem solving and math reasoning, to be less helpful.
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自らを解き放ち生徒の手助けをしません
10:20
And why this is an amazing time to be a math teacher right now
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今数学教師であるのが素晴らしい理由は
10:23
is because we have the tools to create
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上質の教科課程のツールを
10:25
this high-quality curriculum in our front pocket.
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手にしているからです
10:27
It's ubiquitous and fairly cheap,
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至るところにあり、安価です
10:29
and the tools to distribute it
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オープンライセンスで
10:31
freely under open licenses
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自由配布されるこの道具が
10:33
has also never been cheaper or more ubiquitous.
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現在こんなに安く簡単に手に入るのです
10:36
I put a video series on my blog not so long ago
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ブログにビデオシリーズを載せたところ
10:38
and it got 6,000 views in two weeks.
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2週間で6,000件のビューがありました
10:41
I get emails still from teachers in countries I've never visited
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ある国の教師はメールでこう言います
10:44
saying, "Wow, yeah. We had a good conversation about that.
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「素晴らしい。実のある議論になりました
10:47
Oh, and by the way, here's how I made your stuff better,"
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ところで、こう改良したらどうでしょう」
10:50
which, wow.
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素晴らしい提案で
10:52
I put this problem on my blog recently:
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最近ブログにこの問題を載せました
10:54
In a grocery store, which line do you get into,
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店でどのレジの列に入るかです
10:56
the one that has one cart and 19 items
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カート1台に19品の者と
10:58
or the line with four carts and three, five, two and one items.
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4台のカートに3、5、2、1品の者です
11:01
And the linear modeling involved in that was some good stuff for my classroom,
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この線形モデルは教室でも利用でき
11:04
but it eventually got me on "Good Morning America" a few weeks later,
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「グッドモーニング アメリカ」に出演しました
11:07
which is just bizarre, right?
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変ですよね
11:09
And from all of this, I can only conclude
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ここから結論できるのは
11:11
that people, not just students,
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生徒だけではなく、人はー
11:13
are really hungry for this.
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こういうことを切望しているのです
11:15
Math makes sense of the world.
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数学は世の中を理解します
11:17
Math is the vocabulary
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数学は直観力の
11:19
for your own intuition.
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表現手段です
11:21
So I just really encourage you, whatever your stake is in education --
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教育にどんな利害関係があろうとも
11:24
whether you're a student, parent, teacher, policy maker, whatever --
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生徒 保護者 教師 政治家であろうとも
11:27
insist on better math curriculum.
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よりよい教科課程を作成してほしいのです
11:30
We need more patient problem solvers. Thank you. (Applause)
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忍耐強く問題解決する人が必要なのです
ABOUT THE SPEAKER
Dan Meyer - EducatorDan Meyer is exploring the way we teach teachers to teach kids.
Why you should listen
Dan Meyer asks, "How can we design the ideal learning experience for students?" As a part-time Googler, a provocative blogger and a full-time high-school math teacher, his perspective on curriculum design, teacher education and teacher retention is informed by tech trends and online discourse as much as front-line experience with students.
Meyer has spun off his enlightening message -- that teachers "be less helpful" and push their students to formulate the steps to solve math problems -- into a nationwide tour-of-duty on the speaking circuit.
Dan Meyer | Speaker | TED.com