シャドーイング練習: Essence of linear algebra preview - 動画で英語スピーキングを学ぶ
レッスンを作成中...
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Hey everyone!
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So, I'm pretty excited about the next sequence of videos that I'm doing.
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They'll be about linear algebra, which as a lot of you know, is one of those subjects that's required knowledge for just about any technical discipline, but it's also, I've noticed,
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generally poorly understood by students taking it for the first time time.
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A student might go through a class and learn how to compute lots of things, like matrix multiplication, or the determinant, or cross products which use the determinant,
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or eigenvalues, but they might come out without really understanding why matrix multiplication is defined the way that it is,
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why the cross product has anything to do with the determinant, or what an eigenvalue really represents.
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Oftentimes, students end up well-practiced in the numerical operations of matrices, but are only vaguely aware of the geometric intuitions underlying it all.
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But there's a fundamental difference between understanding linear algebra on a numeric level and understanding it on a geometric level.
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Each has its place, but roughly speaking, the geometric understanding is what lets you judge what tools to use to solve specific problems,
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feel why they work, and know how to interpret the results.
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And the numeric understanding is what lets you actually carry through the application of those tools.
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Now, if you learn linear algebra without getting a solid foundation in that geometric understanding, the problems can go unnoticed for a while,
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until you've gone deeper into whatever field you happen to pursue, whether that's computer science, engineering, statistics, economics, or even math itself.
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Once you're in a class, or a job for that matter, that assumes fluency with linear algebra,
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the way that your professors or your co-workers apply that field could seem like utter magic.
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They'll very quickly know what the right tool to use is, and what the answer roughly looks like in a way
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that would seem like computational wizardry if you assume that they're actually crunching all the numbers in their head.
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Here, as an analogy, imagine that when you first learned about the sine function in trigonometry, you were shown this infinite polynomial.
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This, by the way, is how your calculator evaluates the sine function.
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For homework, you might be asked to practice computing approximations of the sine function, by plugging in various numbers to the formula and cutting it off at a reasonable point.
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And in fairness, let's say you had a vague idea that this was supposed to be related to triangles, But exactly how had never really been clear, and was just not the focus of the course.
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Later on, if you took a physics course, where sines and cosines are thrown around left and right, and people are able to tell pretty immediately how to apply them
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and roughly what the sign of a certain value will be, it would be pretty intimidating, wouldn't it?
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It would make it seem like the only people who are cut out for physics are those with computers for brains, and you would feel unduly slow or dumb for taking so long on each problem.
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It's not that different with linear algebra.
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And luckily, just as with trigonometry, there are a handful of intuitions, visual intuitions, underlying much of the subject.
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And unlike the trig example, the connection between the computation and these visual intuitions is typically pretty straightforward.
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And when you digest these, and really understand the relationship between the geometry and the numbers, the details of the subject, as well as how it's used in practice,
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start to feel a lot more reasonable.
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In fairness, most professors do make an effort to convey that geometric understanding.
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The science example is a little extreme.
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But I do think
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that a lot of courses have students spending a disproportionate amount of time on the numerical side of things, especially given that in this day and age,
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we almost always get computers to handle that half, while in practice, humans worry about the conceptual half.
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So this brings me to the upcoming videos.
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The goal is to create a short, binge-watchable series animating those intuitions from the basics of vectors up through the core topics
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that make up the essence of linear algebra.
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I'll put out one video per day for the next five days, then after that put out a new chapter every one to two weeks.
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I think it should go without saying that you cannot learn a full subject with a short series of videos, and that's just not the goal here.
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But what you can do, especially with this subject, is lay down all the right intuitions
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so the learning that you do moving forward is as productive and fruitful as it can be.
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I also hope this can be a resource for educators who are teaching courses that assume fluency with linear algebra, giving them a place to direct students that need a quick brush up.
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I'll do what I can to keep things well paced throughout, but it's hard to simultaneously account for different people's different backgrounds and levels of comfort,
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so I do encourage you to readily pause and ponder if you feel that it's necessary.
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Actually, I'd give that same advice for watching any math video, even if it doesn't feel too quick, since the thinking that you do on your own time is where all the learning really happens, don't you think?
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So, with that as an introduction, I'll see you next video.
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このレッスンの語彙とスピーキングのポイント
このC1レベルのスピーキングレッスンは、動画「Essence of linear algebra preview」を教材にしています。 繰り返し出てくる語は次のとおりです：algebra, linear, understanding, intuitions, geometric。 この動画には、シャドーイング用の文が47文、単語が917語あります。 音声の長さは4:43です。 話す速さは速く、1分あたり約195語です。音のつながりや弱く発音される音が多くなります。 英語の頻出3,000語に含まれる単語は81%だけなので、語彙は難しめです。
この動画の重要語彙
動画の中で特に難しい単語15語を、発音と意味つきで紹介します。
| 単語 | 発音 | 意味 |
|---|---|---|
| algebra 名詞 | /ˈæl.d͡ʒɪ.bɹə/ | 代数 |
| intuition 名詞 | /ˌɪn.tjuːˈɪʃ.ən/ | 直覚, 直感 |
| sine 名詞 | /saɪn/ | サイン, 正弦 |
| determinant 名詞 | /dɪˈtɜːmɪnənt/ | 行列式 |
| compute 動詞 | /kəmˈpjuːt/ | 計算する |
| eigenvalue 名詞 | /ˈaɪ.ɡənˌvæl.ju/ | 固有値 |
| fluency 名詞 | /ˈfluːənsi/ | 流暢 |
| multiplication 名詞 | /ˌmʌltɪplɪˈkeɪʃən/ | 掛け算, 乗法 |
| trigonometry 名詞 | /ˌtɹɪɡəˈnɒmətɹi/ | 三角法 |
| numerical 形容詞 | /n(j)uːˈmɛɹ.ɪ.kəl/ | 数値の |
| computation 名詞 | /ˌkɑmpjʊˈteɪʃn̩/ | 計算 |
| cosine 名詞 | /ˈkoʊ.saɪn/ | コサイン, 余弦 |
| disproportionate 形容詞 | /dɪs.pɹəˈpɔː.ʃə.nət/ | 不釣り合い |
| intimidate 動詞 | /ɪnˈtɪmɪdeɪt/ | 脅す, 威嚇する |
| polynomial 名詞 | /ˌpɑliˈnoʊmiəl/ | 多項式, 整式 |
動画に出てくる句動詞
| 単語 | 発音 | 意味 |
|---|---|---|
| brush up 動詞 | ブラッシュアップ | |
| come out 動詞 | /ˌkʌm ˈaʊt/ | 出る |
| lie down 動詞 | 横になる, 臥す |
この動画の文法
話し手がよく使っている文型を、動画の実際の表現とともに紹介します。
| 文型 | 動画での表現 |
|---|---|
| 受動態 be + 過去分詞 — 誰がするかより、何が起きるかに焦点を当てる | is defined · were shown · might be asked |
| 現在完了形 have/has + 過去分詞 — 過去の出来事が今も関係している | I've noticed · you've gone |
注意したい発音
話し手はI'll, I'm, they'llなど、短縮形や弱形を15回使っています。聞こえたとおりの短い形で発音しましょう。
- 「sh」と「zh」の音: intuition /ˌɪn.tjuːˈɪʃ.ən/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, approximation /əˌpɹɑk.sɪˈmeɪ.ʃən/, computation /ˌkɑmpjʊˈteɪʃn̩/, disproportionate /dɪs.pɹəˈpɔː.ʃə.nət/
- 長い単語（アクセントの位置に注意）: intuition /ˌɪn.tjuːˈɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, eigenvalue /ˈaɪ.ɡənˌvæl.ju/, multiplication /ˌmʌltɪplɪˈkeɪʃən/
日本語話者が苦手な音：
- /r/ と /l/ の区別 — /l/ は舌先を歯茎につけ、/r/ はどこにもつけない: algebra /ˈæl.d͡ʒɪ.bɹə/, numerical /n(j)uːˈmɛɹ.ɪ.kəl/, fruitful /ˈfɹuːtfl̩/, triangle /ˈtɹaɪˌæŋɡəl/, readily /ˈɹɛdɪli/
- /v/ — /b/ にならないように、上の歯を下唇に当てる: eigenvalue /ˈaɪ.ɡənˌvæl.ju/, vaguely /ˈveɪɡli/, convey /kənˈveɪ/, vague /veɪɡ/, vector /ˈvɛktɚ/
- /f/ — 「フ」ではなく、上の歯と下唇で出す: fluency /ˈfluːənsi/, fairness /ˈfɛɚ.nəs/, fruitful /ˈfɹuːtfl̩/, oftentimes /ˈɔf.ənˌtaɪmz/, straightforward /ˌstɹeɪtˈfɔː(ɹ)wə(ɹ)d/
この動画での練習方法
- まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
- まず0.75倍速で一文ずつシャドーイングし、慣れてきたら通常の速度に戻します。
- 自分の声を録音して元の音声と比べます。algebra, intuition, sineなどの単語に特に注意しましょう。
シャドーイングとは？英語上達に効果的な理由
シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。