Практика Shadowing: How I Finally Learned Math (as a Straight-A Engineering Student) - Изучайте разговорный английский по видео

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So in IGCSEs, which is about grade 10, right, which is about a decade ago for me now, 2015, Maths was my least favorite subject.

I hated maths so much.

I was in the lowest level.

and I was failing tests multiple times.

If I even thought about numbers, I felt like throwing up.

I hated maths that much.

But there was one summer where I learned to love maths.

The summer between the two years of IGCSE, I learned to love maths.

And I locked in.

And I ended up doing so well.

that for IB, which is my last two years of high school, I did the most notoriously difficult math subject which is called high -level maths.

And not only that, In engineering, I mean in university, I ended up doing engineering.

And I ended up doing the engineering degree with the most maths,

which is electrical engineering, which is notoriously known as the most difficult engineering degree simply because it has so much maths.

But I was actually so excited for it.

The math subjects were literally my favorite throughout the degree.

And it wasn't because I had like a unique mind for numbers or something

I just learned how to learn maths the proper way And I realized that anybody can enjoy learning maths because

Learning to enjoy learning maths, learning to learn maths, it's all about processes.

Because the thing about maths, the thing about science, is you have to learn them in a very specific way which most students don't know.

Most students approach this subject like they approach all their other subjects.

So here in this video, I'm gonna talk about some principles, which will help you learn maths a lot better that I've learned over the years.

And I'm gonna share my experiences, but also I'm gonna share mainly some lessons that I've learned through a bunch of research that I've done, but also this book that really, really helped me helped me a lot.

called a mind for numbers.

Actually, let me just check.

That's what that book is called A mind for numbers.

Yeah, a mind for numbers by Barbara Oakley, how to excel at maths and science.

So this book literally has some principles that I'm going to go through

that will help you learn maths a lot better and a lot faster because

Maths is probably the subject that is the most hated.

And for good reason, not good reason, but it's the most hated because people don't know how to approach it.

That's the thesis this book was trying to make.

So the first principle, I've got five principles here of learning maths that I'm going to go through.

Okay, so the first one I want to give you is, like I said earlier, you need to understand there's different kinds of subjects.

And depending on the type of subject, they have to be learnt differently.

So I learnt this from the book called Ultra Learning by Scott Young.

He talks about this concept called meta -learning.

where essentially you don't just learn the subject, You need to learn how that specific subject is learnt.

So the book talks about three different types of subjects.

There are procedural type subjects There are, let me just quickly remind myself.

Okay, sorry, there are three things you need to learn: procedures, concepts and facts.

Procedures are like you know, maths procedures, how to solve a maths formula.

Concepts are like, you know, the concept of, you know, the Krebs cycle or something.

And facts are like memorization.

It's like, the mitochondria as the powerhouse of the cell or something.

So there are facts, concepts and procedures that you need to know in every subject.

and different subjects will have different proportions of each.

So, Maths and Science tend to be very heavily procedural -based subjects.

So those are subjects where you need to learn how to do procedures.

You need to learn how to solve integrals.

You need to learn how to do trigonometric solving, that kind of thing.

So procedural subjects are learned very differently to fact -based subjects.

So that's the first thing you need to realize.

You need to realize that maths is learned very, very differently.

to other subjects.

Okay.

So the first principle that I want to give you from this book is very, very interesting.

This book says there are two modes of learning.

Mode number, there's focus mode and there's diffuse mode.

So focus mode is when you're actually sitting there crunching away, trying to solve problems, trying to actually learn the material.

Right?

That's actually intense, you know, problem solving, intense concentration.

But then there's diffuse mode.

which is essentially after you're done doing that, you go and you take a break, right?

You go for a walk, you do whatever.

So this book talks about how you actually need both these modes of learning to actually learn maths.

A lot of the learning that happens in maths doesn't actually happen when you're sitting there crunching away at the numbers.

a lot of the learning actually happens when you step away.

and you take a break, you go for a walk.

So, this is how I like to approach maths.

I like to approach it in chunks.

I like to do like a 20 minute burst of questions.

Take a break.

20 minute burst, take a break.

And I actually learnt this on my own, I figured it out.

I figured it out that when I sat down for like 2 hours and tried to crunch away at maths, I reached the point where I was experiencing diminishing returns.

where I wasn't really learning much more.

And it was actually better for me to take a break.

So this is the big thing I can tell you.

When you take a break and you step away from maths, according to this book, you're still learning.

So it's actually not a bad thing.

So I kind of, I don't really like time myself.

I don't say I'm going to sit here for an hour, I'm going to sit here for two hours.

I kind of just work till my brain starts to feel a bit stuck.

I'll read the textbook till I feel a bit stuck

and either I will jump to a completely different type of question or a completely different that's called chunking,

by the way, where you learn in chunks of similar related material.

So either I'll jump to something else or I'll just take a break for a while.

So you kind of have to sense it out for yourself.

You might sit down for 20 minutes and you'll feel like going into diffuse mode.

But you need to go in and out of these different modes.

and that will help you learn very well.

Thank you.

So, Yeah, it's kind of like, you know how they say like scientists, like often if you do something creative,

for example, a lot of my video ideas They come to me when I'm not even focused.

I'll sit down for two hours and I'll try to come up with video ideas

and nothing will come out of my mind.

But then I'll just like go for a walk or I'll take a shower or something.

And suddenly like 10 video ideas will spawn into my mind.

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So a lot of learning actually happens

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when you take a step away from the books

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and you go for a walk or something and you let the ideas marinate in your mind.

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Right?

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Okay.

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The next principle I want to give you from the book is you have to be careful of what's called Okay, actually, okay, there's a principle I missed here.

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So when it comes to those procedural type subjects, like I said...

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a lot of the learning, you have to learn by solving problems.

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There's no way around it.

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A procedural based subject is a subject where you see a brand new problem that you've never seen before.

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and you need to be able to solve it from scratch.

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And there's no way of being able to do that except for experiencing so many different scenarios

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that you're able to solve that problem.

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I kind of think of tennis as a procedural based subject, right?

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So, you can practice as much as you want drills, right?

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Like I'm going to hit the ball in this specific place.

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And you get you can get very good at aiming the ball in a very specific place.

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but an opponent is going to be unpredictable.

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you're never going to know what the opponent's going to do.

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You're never going to know how fast he's going to hit the ball, at what angle, at what speed, at what...

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at what time of day, at what height, whatever.

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So the only way to get good at tennis is by practicing

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so much that you're able to handle those kind of enemies.

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So same thing with maths.

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you need to prioritize doing practice problems.

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So in HL Maths, like I said before, I actually didn't get the top mark.

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It was an insanely difficult subject.

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I actually got the second to highest mark.

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But the thing is, I had basically memorized the textbook at that point.

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I had basically memorized every paragraph of the textbook.

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I understood the concepts perfectly.

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I had memorized like the mark schemes and everything.

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But the reason I didn't do the best was simply I had not practiced enough.

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I had access to this huge stack of past papers, right?

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Thank you.

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And I only did like three quarters of them.

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If I had have completed that stack, I probably would have gotten the top mark.

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So here I am, I understood the concepts perfectly, I could recite the textbook from my mind.

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And I didn't do well because I didn't practice enough.

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So you have to go through the procedures.

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You have to learn by doing active recall, which is you know, countless times tested and proven scientifically to be the most effective study technique,

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especially for procedural type subjects, you need to do a ton of practice questions till it becomes easy.

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Okay, this brings me to principle three.

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from the book.

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This is a very interesting one.

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It talks about the illusion of competence. So, Sometimes...

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we feel like we understand something in maths, But we don't.

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So I had a friend, right?

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I don't know why I'm telling this story, but I had a friend in first semester, first year of university.

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He was studying calculus all day long.

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He was binge watching the lectures.

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He was writing notes.

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I thought this guy was going to get, you know, the top marks.

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He was rewatching the lectures over and over again.

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He knew the material deeply.

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- Thank you.

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He ended up failing that subject.

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Now, kind of funny looking back now, but we can learn from him from his mistakes.

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The reason he failed was because He was under the illusion that he had learnt the material.

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But he hadn't.

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It's kind of like if you binge watch videos about playing tennis, You can get very good at understanding tennis and you think you know how to play tennis.

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But as soon as you get onto the court and you try to hit a ball, You're gonna miss completely because you haven't actually practiced.

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You haven't gotten direct experience.

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So the book Ultra Learning that I talked about earlier, he gives this principle called directness, where you actually need direct experience with the thing to learn it.

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So in maths, that's through doing problems, that's through actually crunching your brain.

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I will sometimes memorize an entire chapter, memorize the worked problems, memorize all that.

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And as soon as I try to do a problem, I can't do it.

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Because you need the direct experience.

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So be careful of this thing called the illusion of competence, where you feel like you understand the concepts.

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So seeing solutions doesn't mean you understand.

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Rereading the textbook and feeling like things look familiar doesn't mean you understand.

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Um...

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because we're afraid of this.

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But the thing is, we don't do the problems.

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because we're afraid of destroying our confidence.

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Rereading the textbook, makes us feel good.

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It's like, "Oh, I understand this.

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This is good.

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I'm going to do so well." But doing a problem and failing it Feels bad.

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That's why we avoid it, because we take a hit to our ego.

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And so this is the next principle that I want to give you, which I think probably is probably one of the most important principles here.

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But failure is a necessary part of the learning process.

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It's not an optional thing.

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- Okay.

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Like many students, this is what they do.

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They think, "Okay, I'm gonna do practice problems, I'm gonna do past papers," once I understand the material well.

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And once I understand the material well, I'm going to get all the questions right and I'm going to do so well.

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They use the past papers to see if they understand the material.

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rather than to actually use it to learn in the first place.

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So for me, past papers are for learning.

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So what I've got here is We're afraid of destroying our confidence, but failure is so important.

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Because failure signals to our brain that it's time to learn.

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where you get something wrong, It tells your brain that what you're doing right now is important.

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When you go for a tennis swing and you miss, A really good way, by the way, to learn about how failure is so important is learn how to juggle.

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So what juggling is?

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I don't have juggling balls with me right now.

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But juggling is basically where you have three balls and you essentially have one go over the top.

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You have one ball go over the top, and at all times, and you just kind of throw them back and forth like that.

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And it takes about three hours to learn how to juggle.

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but you will drop the balls like a thousand times.

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But every time you drop the ball, If you're a weak -minded person, you'll drop the ball and you'll be like, oh, I failed to juggle, oh, no, man.

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And you'll give up.

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But if you understand the science of learning, you'll realize that every time you drop the ball, It's signaling to your brain to make a change.

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Like, okay, it doesn't happen consciously, by the way.

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Your brain is very, very smart.

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Your brain does all the learning behind the scenes.

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So it's like, okay, I dropped the ball.

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And your brain will go, okay, we dropped the ball, we're going to throw it differently this time, blah, blah, blah.

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and you fail a thousand times and you will learn to juggle.

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It'll take like three hours.

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But the same thing is with maths.

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You fail questions enough times, you will learn how to do it.

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So like I said, you can memorize the textbook, but as soon as you begin getting, doing questions, you're going to get them wrong.

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And here's the difference between top and bottom students.

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Bottom students see getting a question wrong, they see failure.

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as proof that they're dumb.

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"Oh man, I dropped the ball, I'm dumb, I can't juggle." versus I dropped the ball top students They see it as feedback.

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Okay, I got this question wrong.

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What can I learn from it?

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What's some feedback I can get?

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So, Basically, another way to put it from the book that's very interesting is failure is your edge of understanding.

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So maybe right now in maths you've got this small area of understanding.

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when you have to learn this entire area of maths.

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Where you're getting questions wrong, that is your edge of understanding.

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So you need to face that and you need to learn from your mistakes

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so you can get bigger and bigger and eventually encompass the entire area of maths.

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So the biggest tip I can give you here that I've learned is keep a failure log.

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So track all your failures.

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So I had a friend, he was doing past papers.

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a couple semesters ago.

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And Every time he would get a question wrong, He would delete that question from, he would have like past papers.

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When he would get a question wrong, he would like cross out the question

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because he never wanted to see it again because getting the question wrong made him feel bad.

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And that's not the correct way to approach it.

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The best way to approach it, actually, what I like to do, is if I get a question wrong, I take out a red coloured pen and I write what I did wrong and I correct myself.

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and I learn from the mistake.

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And I come back and I constantly review those errors that I made.

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So if you're doing it in your notebook, you can do it in red.

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So you constantly know to come back to the red areas.

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So work on your weaknesses.

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for if you're learning any skill right, you want to work on your weaknesses.

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If you're trying to get better at tennis, you want to like maybe, you know, your backswing is bad.

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You want to work on your backswing. So...

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Focus on your weaknesses and that's how you're going to learn.

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So keep a failure log.

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keep coming back to it.

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- Thank you.

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Okay, the next principle I want to give you is a principle called chunking.

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Thank you.

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So I guess I talked about this earlier, but basically you want to learn maths in chunks of different concepts.

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So let's say for example you're learning integration.

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Integration is nothing but You need to know some multiplication, you need to know like exponent rules, you need to know trigonometric rules,

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you need to know all these things and finally you need to know integration.

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So you need to learn these things in chunks, if that makes sense.

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So chunks are like small groups of patterns that our brain associates to be one similar thing.

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So what I like to do is I like to learn chunks with spaced repetition.

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So I might spend 20 minutes trying to understand this one concept.

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and I might jump to a different chunk and spend 20 minutes trying to learn that concept.

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then I'll jump to a different chunk and try to learn a different concept.

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So the other thing is spaced repetition, by the way you can't The idea of spaced repetition is Our brains forget fast.

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You learn something and you'll forget it within a day.

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But if you review that material right when your brain is about to forget, your brain will actually learn it a bit stronger and it'll take it longer to forget.

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So with maths, you really need to spread out your learning.

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it's very difficult to cram maths on the day before the exam.

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Like, it's possible, but it's very difficult because you need to give your brain some space to consolidate the learning.

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So I like to learn chunks and then go away

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and come back to it and go away and come back to it.

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I like to attack maths with these bursts.

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of energy, kind of like you're doing a strafe run, a bombing run.

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I'll do the maths for a little bit and I'll come back.

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And I'll do it a little bit and I'll come back.

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I'll do it a little bit and I'll come back.

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So we can actually connect that with the diffuse mode that I talked about earlier.

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Sometimes when I'm doing, you know, especially in the exams, if I don't know how to do a question, There's something from the book that it mentions.

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Let me see if I can...

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I don't know how to pronounce it.

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It's called the Einstellung -Effekt.

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The Einstein effect is that we cling to the first approach to doing a question.

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So actually sometimes the first approach we cling to doing a question isn't the best approach to actually learn it, to actually do the question.

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So what I like to do when I'm doing an exam for example is

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if I don't know how to do a question, I will literally just go away and I'll do other questions And often that activates the diffuse mode I talked about earlier,

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where you let your brain marinate on the problem.

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Often I'll come back and I'll think of a different way to approach the problem.

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So often we get attached to the first way to solve a problem

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But actually it can be good to actually step away and let our brain try different ways to solve the problems.

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- Thank you.

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Okay, so the final tip I want to give you here is my own one.

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but use different resources.

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So when I was learning, let's say for example, okay, linear algebra, it's this subject about like vectors and stuff multiple dimensional vectors.

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When I was learning linear algebra, in university.

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I found this playlist on YouTube that was so good called the essence of linear algebra.

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And basically this playlist explained linear algebra in a very entertaining visual way.

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And the thing is, linear algebra is, like many maths subjects, it's a very abstract concept.

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We're talking about vectors in like 10 dimensions and stuff.

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It's very difficult to actually understand what you're learning.

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And so I found it very painful because if you don't know, the best way to actually learn maths is to actually understand it.

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If you understand why a formula is why it is, you will learn it way better than if you just simply memorize the formula, if you just simply memorize the steps to solve it.

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So if you can actually understand why it is how it is, you will learn a lot better.

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So I would binge watch this series on YouTube called

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the essence of linear algebra by a three brown one blue I guess.

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I would literally like, a bunch of us would actually like put it onto the projector and we'd watch it together.

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Because I binge watched that series and it really, really, first of all, it helped me love that subject because I was actually understanding the material.

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I wasn't just brute memorizing.

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I wasn't just brute memorizing.

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I was actually understanding the material.

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But second of all, it helped me learn it a lot better because I actually understood it.

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If you focus on understanding, you will learn a lot better.

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So, use multiple resources.

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In high school, I used multiple textbooks.

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I used YouTube.

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I used friends.

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I used teachers.

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Get all these different perspectives to learn the material.

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and it will help you a lot more.

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It will also inject curiosity and excitement into you, which will make it more fun to learn.

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I literally loved learning maths.

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and a big reason was because some of these teachers made learning so exciting.

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It made maths so exciting.

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If I just stuck to like a boring teacher that I had, I probably wouldn't have enjoyed Mass that much.

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So, Yeah, go enjoy.

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Maths is literally like such a beautiful thing to learn.

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It's like a whole world to explore.

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There's so much history.

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If you learn about some of the history of some of the mathematicians, It is super interesting.

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like um one of the funniest mathematicians that has some interesting stories i think pierre de fermat Pierre de Fermat, I think that's his name.

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He has some hilarious stories.

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But yeah, go learn the stories of mathematics.

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Go enjoy mathematics.

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It's actually a lot of fun.

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If you think you suck at maths, Don't trust that thought.

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It's not real.

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It was implanted to you long ago because nobody knows how to study maths properly.

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Once you learn how to study maths properly, I guarantee it's a lot of fun.

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So I'm just thinking, are there any other tips I can give you for the exam?

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for actually sitting the exam.

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No, yeah, like when I'm actually sitting the exam, I will just come back to questions like I said, because my mind will work on them in the background. do different

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you know yeah that's pretty much it so um go go ahead

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and study maths let me know if this video helped you below But apart from that, go watch this video now.

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But yeah, see you later.

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Goodbye.

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Farewell.

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Yeah, and learn how to juggle.

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Don't forget.

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Learn how to juggle.

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Fuck.

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Like this.

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You just throw these items in the air.

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Bye.

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It's very difficult to juggle something that's floating in the air.

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Anyway, see you later.

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Этот урок разговорной практики уровня B2 построен на видео «How I Finally Learned Math». Чаще всего повторяются слова: learn, math, learning, subject. В этом видео 349 предложений и 4156 слов для шедоуинга. Речь длится 21:35. Говорящий говорит быстро, около 192 слов в минуту, поэтому звуки часто сливаются и сокращаются. 83% слов входят в 3000 самых частых слов английского языка; остальные стоит разобрать перед занятием.

Ключевая лексика этого видео

Самые сложные слова из видео (15), с произношением и значением:

Слово	Произношение	Значение
mathematics существительное	/mæθ(.ə)ˈmæt.ɪks/	матема́тика
memorize глагол	/ˈmɛm.əˌɹaɪ̯z/	запомина́ть, запо́мнить
chunk существительное	/t͡ʃʌŋk/	кусо́к
juggle глагол	/ˈd͡ʒʌɡəl/	жонгли́ровать
textbook существительное	/ˈtɛkst.bʊk/	уче́бник, посо́бие
procedural прилагательное	/pɹəˈsiːd͡ʒ(ə)ɹəl/	процеду́рный
algebra существительное	/ˈæl.d͡ʒɪ.bɹə/	а́лгебра
linear прилагательное	/ˈlɪn.i.əɹ/	лине́йный
diffuse глагол	/dɪˈfjuz/	рассе́ивать, рассе́ять
binge существительное	/bɪnd͡ʒ/	пиру́шка, запо́й
crunch глагол	/kɹʌnt͡ʃ/	хрусте́ть, хру́стнуть
juggling существительное	/ˈd͡ʒʌɡəlɪŋ/	жонгли́рование, жонглёрство
repetition существительное	/ˌɹɛp.əˈtɪʃ.ən/	повторе́ние
illusion существительное	/ɪˈl(j)uːʒ(ə)n/	иллю́зия
vector существительное	/ˈvɛktɚ/	ве́ктор

Фразовые глаголы, которые вы услышите

Слово	Произношение	Значение
come back глагол		возвраща́ться, возврати́ться
end up глагол		зака́нчиваться, зако́нчиться
go away глагол		уходи́ть, уйти́
go over глагол		хорошо́ принима́ться
come up with глагол		придумывать, приходить в голову
cross out глагол		вычёркивать, вы́черкнуть
give up глагол		сдава́ться, сда́ться
go ahead глагол	/ˌɡoʊ əˈhɛd/	дава́й, приступа́йте

Фразы, которые стоит повторять

Короткие законченные фразы из видео, которые пригодятся в повседневном разговоре:

That's actually intense, you know, problem solving, intense concentration.
You haven't gotten direct experience.
I don't have juggling balls with me right now.
What can I learn from it?
What's some feedback I can get?

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Конструкции, которые говорящий использует чаще всего, с точными словами из видео:

Конструкция	В видео
Условные предложения if + предложение, will/would + глагол — условие и его результат	If you're a weak -minded person, you'll drop · If you focus on understanding, you will learn
Пассивный залог be + причастие прошедшего времени — важно, что происходит, а не кто это делает	is called · be learnt · is learnt
Present Perfect have/has + причастие прошедшего времени — прошлое действие, важное сейчас	I've learned · you've never seen · have completed
Придаточные определительные who / which + предложение — уточнение о человеке или предмете	IGCSEs, which is · subject which is · engineering, which is

Произношение, на которое стоит обратить внимание

Говорящий использует 121 сокращённых и редуцированных форм, например I'm, you're, I'll. Произносите их коротко, так, как слышите.

Звуки «th»: mathematics /mæθ(.ə)ˈmæt.ɪks/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, thesis /ˈθisɪs/
Звуки «sh» и «zh»: repetition /ˌɹɛp.əˈtɪʃ.ən/, illusion /ɪˈl(j)uːʒ(ə)n/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, diminish /dɪˈmɪnɪʃ/, memorization /ˈmem(ə)ɹaɪˌzeɪʃn̩/
Длинные слова — следите за ударением: mathematics /mæθ(.ə)ˈmæt.ɪks/, procedural /pɹəˈsiːd͡ʒ(ə)ɹəl/, repetition /ˌɹɛp.əˈtɪʃ.ən/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, notoriously /nə(ʊ)ˈtɔː.ɹɪ.əs.li/

Звуки, трудные для русскоязычных:

/w/ — губы округлены, это не /v/: weakness /ˈwik.nɪs/, rewatch /ɹiːˈwɒtʃ/, farewell /fɛɹˈwɛl/
Звонкие согласные в конце слова — не оглушайте их: memorize /ˈmɛm.əˌɹaɪ̯z/, diffuse /dɪˈfjuz/, binge /bɪnd͡ʒ/, reread /ɹiːˈɹiːd/, prioritize /pɹaɪˈɒɹ.ə.taɪz/
/ŋ/ в конце слова — носовой звук без «г»: juggling /ˈd͡ʒʌɡəlɪŋ/, cling /ˈklɪŋ/

Как заниматься с этим видео

Прослушайте всё видео один раз молча и выпишите незнакомые слова.
Начните со скорости 0,75×, повторяйте предложение за предложением и вернитесь к обычной скорости, когда станет легко.
Запишите себя и сравните с оригиналом, обращая внимание на такие слова, как mathematics, memorize, chunk.

Что такое техника Shadowing?

Shadowing — это научно обоснованная техника изучения языка, изначально разработанная для подготовки профессиональных переводчиков и популяризированная полиглотом доктором Александром Аргуэльесом. Метод прост, но эффективен: вы слушаете аудио на английском от носителей языка и немедленно повторяете вслух — как тень, следующая за говорящим с задержкой в 1–2 секунды. В отличие от пассивного прослушивания или грамматических упражнений, Shadowing заставляет мозг и мышцы рта одновременно обрабатывать и воспроизводить реальные речевые паттерны. Исследования показывают, что это значительно улучшает точность произношения, интонацию, ритм, связную речь, понимание на слух и беглость речи — что делает его одним из самых эффективных методов для подготовки к IELTS Speaking и реального общения на английском.

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