Shadowing Practice: Essence of linear algebra preview - Learn English Speaking with Video
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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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Vocabulario y notas de pronunciación para esta lección
Esta lección de conversación de nivel C1 se basa en el vídeo “Essence of linear algebra preview”. Las palabras que más se repiten: algebra, linear, understanding, intuitions, geometric. Este vídeo tiene 47 frases y 917 palabras para practicar shadowing. La parte hablada dura 4:43. El hablante habla rápido, unas 195 palabras por minuto, así que espera sonidos enlazados y reducidos. Solo el 81 % de las palabras está entre las 3.000 más comunes del inglés, por lo que el vocabulario es exigente.
Vocabulario clave de este vídeo
Las 15 palabras más avanzadas del vídeo, con su pronunciación y significado:
| Palabra | Pronunciación | Significado |
|---|---|---|
| algebra sustantivo | /ˈæl.d͡ʒɪ.bɹə/ | álgebra |
| intuition sustantivo | /ˌɪn.tjuːˈɪʃ.ən/ | intuición |
| geometric adjetivo | /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/ | geométrico |
| sine sustantivo | /saɪn/ | seno |
| determinant sustantivo | /dɪˈtɜːmɪnənt/ | determinante |
| matrix sustantivo | /ˈmeɪ.tɹɪks/ | matriz |
| compute verbo | /kəmˈpjuːt/ | computar, calcular |
| eigenvalue sustantivo | /ˈaɪ.ɡənˌvæl.ju/ | autovalor, valor propio |
| fluency sustantivo | /ˈfluːənsi/ | fluidez, fluencia |
| multiplication sustantivo | /ˌmʌltɪplɪˈkeɪʃən/ | multiplicación |
| numeric adjetivo | /n(j)uːˈmɛɹɪk/ | numérico |
| trigonometry sustantivo | /ˌtɹɪɡəˈnɒmətɹi/ | trigonometría |
| numerical adjetivo | /n(j)uːˈmɛɹ.ɪ.kəl/ | guarismo |
| fairness sustantivo | /ˈfɛɚ.nəs/ | imparcialidad, equidad |
| animate adjetivo | /ˈæn.ə.mət/ | animado |
Phrasal verbs que vas a escuchar
| Palabra | Pronunciación | Significado |
|---|---|---|
| put out verbo | sacar | |
| brush up verbo | desempolvar, bruñir | |
| carry through verbo | llevar a término, llevar a cabo | |
| come out verbo | /ˌkʌm ˈaʊt/ | revelarse, salir a la luz |
| end up verbo | terminar, ir a parar | |
| go through verbo | atravesar | |
| lie down verbo | echarse, tenderse | |
| make up verbo | /ˌmeɪk ˈʌp/ | reponer, suplir |
Gramática en este vídeo
Las estructuras que más usa el hablante, con las palabras exactas del vídeo:
| Estructura | En el vídeo |
|---|---|
| Voz pasiva be + participio pasado — importa lo que ocurre, no quién lo hace | is defined · were shown · might be asked |
| Present perfect have/has + participio pasado — una acción pasada que sigue importando ahora | I've noticed · you've gone |
Pronunciación a tener en cuenta
El hablante usa 15 contracciones y formas reducidas, como I'll, I'm, they'll. Dilas en su forma corta, tal como las oyes.
- Los sonidos de “sh” y “zh”: intuition /ˌɪn.tjuːˈɪʃ.ən/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, approximation /əˌpɹɑk.sɪˈmeɪ.ʃən/, computation /ˌkɑmpjʊˈteɪʃn̩/, disproportionate /dɪs.pɹəˈpɔː.ʃə.nət/
- Palabras largas — cuida el acento: intuition /ˌɪn.tjuːˈɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, eigenvalue /ˈaɪ.ɡənˌvæl.ju/, multiplication /ˌmʌltɪplɪˈkeɪʃən/
Sonidos difíciles para hispanohablantes:
- /v/ — no es /b/: los dientes tocan el labio inferior: eigenvalue /ˈaɪ.ɡənˌvæl.ju/, vaguely /ˈveɪɡli/, convey /kənˈveɪ/, vague /veɪɡ/, vector /ˈvɛktɚ/
- /z/ — sonora, no /s/: oftentimes /ˈɔf.ənˌtaɪmz/, wizardry /ˈwɪzədɹi/, pause /pɔːz/
- /dʒ/ — no es la “y” ni la “ch”: algebra /ˈæl.d͡ʒɪ.bɹə/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, educator /ˈɛd͡ʒəkeɪtɚ/, binge /bɪnd͡ʒ/, digest /daɪˈd͡ʒɛst/
Cómo practicar con este vídeo
- Escucha el vídeo entero una vez sin hablar y anota las palabras que no conoces.
- Empieza a velocidad 0,75×, haz shadowing frase por frase y vuelve a la velocidad normal cuando te resulte fácil.
- Grábate y compara con el original, prestando atención a palabras como algebra, intuition, geometric.
¿Qué es la Técnica de Shadowing?
Shadowing es una técnica de aprendizaje de idiomas respaldada por la ciencia, desarrollada originalmente para la formación de intérpretes profesionales y popularizada por el políglota Dr. Alexander Arguelles. El método es simple pero poderoso: escuchas audio en inglés nativo y lo repites en voz alta de inmediato, como una sombra que sigue al hablante con solo 1-2 segundos de retraso. A diferencia de la escucha pasiva o los ejercicios de gramática, el shadowing obliga a tu cerebro y músculos de la boca a procesar y reproducir simultáneamente patrones de habla reales. Las investigaciones muestran que mejora significativamente la precisión de la pronunciación, la entonación, el ritmo, el habla conectada, la comprensión auditiva y la fluidez al hablar, convirtiéndola en una de las metodologías más efectivas para la preparación del IELTS Speaking y la comunicación en inglés en el mundo real.