Pratique du Shadowing: Essence of linear algebra preview - Apprendre l'anglais à l'oral avec la vidéo
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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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Vocabulaire et conseils d’expression pour cette leçon
Cette leçon d’expression orale de niveau C1 s’appuie sur la vidéo « Essence of linear algebra preview ». Les mots qui reviennent le plus souvent : algebra, linear, understanding, intuitions, geometric. Cette vidéo contient 47 phrases et 917 mots à répéter en shadowing. La partie parlée dure 4:43. Le locuteur parle vite, environ 195 mots par minute : attendez-vous à des liaisons et à des sons réduits. Seuls 81 % des mots font partie des 3 000 mots les plus courants en anglais, le vocabulaire est donc exigeant.
Vocabulaire clé de cette vidéo
Les 15 mots les plus avancés de la vidéo, avec leur prononciation et leur sens :
| Mot | Prononciation | Sens |
|---|---|---|
| algebra nom | /ˈæl.d͡ʒɪ.bɹə/ | algèbre |
| intuition nom | /ˌɪn.tjuːˈɪʃ.ən/ | intuition |
| geometric adjectif | /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/ | géométrique |
| sine nom | /saɪn/ | sinus |
| determinant nom | /dɪˈtɜːmɪnənt/ | déterminant |
| compute verbe | /kəmˈpjuːt/ | computer, calculer |
| eigenvalue nom | /ˈaɪ.ɡənˌvæl.ju/ | valeur propre |
| multiplication nom | /ˌmʌltɪplɪˈkeɪʃən/ | multiplication |
| trigonometry nom | /ˌtɹɪɡəˈnɒmətɹi/ | trigonométrie |
| numerical adjectif | /n(j)uːˈmɛɹ.ɪ.kəl/ | numérique |
| fairness nom | /ˈfɛɚ.nəs/ | justice |
| animate adjectif | /ˈæn.ə.mət/ | animé |
| approximation nom | /əˌpɹɑk.sɪˈmeɪ.ʃən/ | rapprochement |
| computation nom | /ˌkɑmpjʊˈteɪʃn̩/ | calcul, comptage |
| cosine nom | /ˈkoʊ.saɪn/ | cosinus |
Les verbes à particule que vous entendrez
| Mot | Prononciation | Sens |
|---|---|---|
| brush up verbe | dérouiller | |
| come out verbe | /ˌkʌm ˈaʊt/ | sortir |
| end up verbe | se terminer, se finir | |
| go through verbe | traverser | |
| lie down verbe | s'étendre, se coucher | |
| make up verbe | /ˌmeɪk ˈʌp/ | constituer, composer |
La grammaire de cette vidéo
Les structures que le locuteur utilise le plus, avec les mots exacts de la vidéo :
| Structure | Dans la vidéo |
|---|---|
| Voix passive be + participe passé — l’accent est mis sur ce qui arrive, pas sur qui le fait | is defined · were shown · might be asked |
| Present perfect have/has + participe passé — une action passée qui compte encore maintenant | I've noticed · you've gone |
Prononciation à surveiller
Le locuteur utilise 15 contractions et formes réduites, comme I'll, I'm, they'll. Prononcez-les sous leur forme courte, telles que vous les entendez.
- Les sons « sh » et « zh »: intuition /ˌɪn.tjuːˈɪʃ.ən/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, approximation /əˌpɹɑk.sɪˈmeɪ.ʃən/, computation /ˌkɑmpjʊˈteɪʃn̩/, disproportionate /dɪs.pɹəˈpɔː.ʃə.nət/
- Mots longs — placez bien l’accent: intuition /ˌɪn.tjuːˈɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, eigenvalue /ˈaɪ.ɡənˌvæl.ju/, multiplication /ˌmʌltɪplɪˈkeɪʃən/
Les sons difficiles pour les francophones :
- /tʃ/ et /dʒ/ — à ne pas adoucir en « ch » et « j »: algebra /ˈæl.d͡ʒɪ.bɹə/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, educator /ˈɛd͡ʒəkeɪtɚ/, binge /bɪnd͡ʒ/, crunch /kɹʌnt͡ʃ/
- /r/ anglais — langue recourbée, sans frotter la gorge: algebra /ˈæl.d͡ʒɪ.bɹə/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, matrix /ˈmeɪ.tɹɪks/, numeric /n(j)uːˈmɛɹɪk/, trigonometry /ˌtɹɪɡəˈnɒmətɹi/
Comment s’entraîner avec cette vidéo
- Écoutez la vidéo en entier une fois sans parler et notez les mots que vous ne connaissez pas.
- Commencez à la vitesse 0,75×, répétez phrase par phrase, puis revenez à la vitesse normale quand cela devient facile.
- Enregistrez-vous et comparez avec l’original, en faisant attention à des mots comme algebra, intuition, geometric.
Qu'est-ce que la technique du Shadowing ?
Le Shadowing est une technique d'apprentissage des langues fondée sur la science, développée à l'origine pour la formation des interprètes professionnels. Le principe est simple mais puissant : vous écoutez de l'anglais natif et le répétez immédiatement à voix haute — comme une ombre suivant le locuteur avec un décalage de 1 à 2 secondes. Les recherches montrent une amélioration significative de la précision de la prononciation, de l'intonation, du rythme, des liaisons, de la compréhension orale et de la fluidité.