Shadowing Practice: We Let $200 GPT-Pro Think for 7 Days on Unsolved Math - Learn English Speaking with Video

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So as some of you will have seen, the rate at which AI is solving research level maths problems has accelerated drastically in the last few months.

Problem after problem after problem is supposedly being solved by these large AI teams like OpenAI and Anthropic.

You know, recently we saw that a larger proportion of the zeros of the Riem

and Zeta function have been proven by Claude to lie on the critical line, which is a development of a long line of work, one of the main ones being Brian Connery's work,

who showed that an initial proportion of the zeros of the Riem and Zeta function lie on this critical line.

And there's a really funny quote from a talk that I saw once.

In 1989, I proved that at least two-fifths of the zeros are on the line.

When they offered this million-dollar prize, I wrote to them and said, well, I've already proven two-fifths of this, so I think you should give me $400,000.

But the thing is, the majority of these really difficult problems have been solved with harnesses

or models which haven't been released to the public yet.

Nonetheless, in the last few videos, we've seen some serious improvements in the performance of these models, solving PhD kinds of problems that I've come across in my research,

and today we're going to be taking it, like, another step further.

Because, like I mentioned last time, a member from OpenAI's codex team reached out to me

and basically planned a meeting where he explained to me how

I might be able to tackle some of these problems using codex in a more kind of refined and systematic way.

Since then, I've also had a member of Anthropic reach out, so I've actually now got both of the best models, like the $200 subscriptions, which is absolutely awesome, so thank you so much to both of you guys,

and they've even said to me, you don't have to be biased, so honestly that's just really awesome.

But in any case, today we're going to be using this in order to have $200

chat GPT think for well over a week on a class of open mathematical problems in random matrix theory

that would be really cool to see an answer to, even if it's an answer that's written in clanker language, okay?

Because one of the problems with having such powerful models is the more powerful they become, the harder it is to kind of interpret in a human way what exactly it is that they're doing.

We saw that last time

when we gave Anthropic $100 of AI tokens to see whether it could reproduce a research file that has any interesting results.

And honestly, it felt a bit like I'd given Dario $100 and then the client call was like, $100 please.

It took the money, said a bunch of gibberish.

It was actually kind of true.

So it's not that it did things wrong, but maybe it didn't actually give the types of results that I thought were interesting.

But in the last week, I've had codecs running nonstop.

There's been lines of reasoning that have gone on for one day, lines of reasoning that have gone on for two days.

We're well over seven days now of the clanker thinking in

understanding something that's quite important with regards to the Riemann zeta function, but that's actually approached from a completely different angle using tools of random matrix theory.

So I'm going to be discussing basically how this system works, and we're going to be looking like kind of inside the mind of the clanker that's been thinking for ages, because it does feel a bit like it's been in some long trance.

I was reading the reasoning, but generally speaking, it was quite hard actually to understand what exactly it was doing.

But anyway, we're going to load up the project now, and I'm going to show you exactly how it's been done.

Let's go.

So as you can see, we're using the Mac application for codecs, and actually what it's been doing is it's been working in this kind of subfolder, but as you can see here, we've been doing a whole bunch of research,

basically starting from this $100 Clawed tokens that we gave it last time, and I've been continuing this essentially with a whole bunch of MD files.

The Clankers are trying loads of different things, I mean, as you can see, it's doing loads of like tracking of the things it's tried, where it's gone.

I'm basically just trying to collect with all of the tokens that I've been given access to through both companies, all of the different paths.

So to try and search to see if there is a way of doing it.

This is some study of the modes of the distribution that I'm trying to understand.

At this point, I'm just trying to see whether the clanker can agree

that it understands something and then to try and systematically read that path that it does, as opposed to reading the thousands of things that it's done.

I mean, just look at all of these files.

It's just ridiculous.

There's just no way I can read this at this point.

So I'm taking the method a bit to the extreme compared to what we've done in the last few years, because, you know, we've seen these models have become much, much better at doing these types of tasks,

have it go off in all of the directions.

And then when it finds a direction that's useful for me to then follow the reasoning of that direction, to have the models verify their reasoning before.

So this is the chat after Codex has crashed.

It was thinking for about 24 hours and it had edited loads of files

and added them to the git folder

so all of its chain of thought was basically in editing these .md files

and I asked it to essentially summarize what it had done.

So there it took three minutes.

So this number, just looking at it, is quite interesting because I think what it's basically doing is positioning the mode

and it's giving you an actual location for it.

As you can see, we kind of had it think for way longer

and it was giving all kinds of explanations of what exactly was going on.

So the problem that I was basically asking it to look at, which is to do with why a certain distribution, as you go up in matrix dimension, there's like a transition where it goes from having one mode to having two bumps.

And the thing that most people are interested in is whether it still has two bumps when the dimension goes to infinity.

After reading this for quite a while, so there we've got another hour, there we've got 17 minutes, which is nothing, then we've got one minute, there we've got 13 hours of it basically working on this problem,

it started to feel like what it was basically giving was like some sort of machine proof.

So it was essentially incomprehensible to me.

So it gave all of these summaries, so this is still its reasoning, still its reasoning.

It was literally just thinking for ages.

This is the 13 hours of chain of thought.

So I'm just going to scroll down.

It just did loads of stuff, okay?

This is what I mean.

It's kind of just, I don't know whether it's just hallucinating.

I don't think, I think there is a lot of truth in what it's doing, but to me, I can't, it's just not possible, okay, for a human to read through all of this stuff.

Then I asked it for a recap after this 13 hours of thinking, which was a lot.

Basically it was just giving me loads of numerical numbers, like, and then it basically says that they're not actually numerical, but they're kind of like from inequalities,

basically saying that it was a form of computer assisted proof.

Yeah, so current work is largely rigorous computer assisted analysis, not merely numerical experimentation now.

Essentially it was taking polynomials and trying to understand, write them in a certain basis, and then showing that the coefficients like have a certain sign,

and then that can kind of deduce information about the question that we're asking to do with these modes.

That's my understanding of it, okay?

But nonetheless, it kind of summarized something quite interesting, which is this.

So this is a general n expression of something that you need to evaluate.

It did manage to come up with a candidate justification as to why n equals 9 was special.

But then it couldn't really prove rigorously that that was actually the case.

Kind of said that I was not really that interested in all of these kind of machine-assisted proofs, I really just wanted like an analytic description of what was different at n equals 9 compared to the other ones.

Here I was just trying to set up a plan, so this is a bit like where you can essentially just ask Codex to set up a plans.md file

or some sort of like systematic way, this was guidance that was given from the guy from Codex, but you have a discussion with the model in order to get this done, so here you see it had like a research tracker

and then it had this like analytic program

which is where it actually has a bunch of tasks that it's going to try and solve.

So then it basically outlined which things are possible, so medium to low probability that it could prove that there's two maxima in the limit, which is kind of one of the most interesting problems.

So then it summarized all of the known results, and this is where it started to work on thinking for a very very long time, and it's literally been thinking for multiple days now,

like, and we're going to see what it's done.

So this is the document that the Clankers have produced, which is summarizing this large hallucination, which is probably actually legit,

I am only joking, but it is funny seeing so much stuff be generated.

So we're gonna go through it.

It's made a 16-page document with a final assessment.

All hell breaks loose when you just let the clankers do it, but maybe that's how you should do it.

We don't know yet.

So it's worth trying.

But okay, so it's basically just outlining the research tracker.

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So you can see it's got 321 markdown research notes.

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It's just made loads of stuff.

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Two different logarithmic maxima.

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The density therefore is generated by the switching set for at least two.. oh my goodness this doesn't make any sense.

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The one constraint formula.

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Okay, so this is cool.

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In principle, if I understand it right, which I'm not even sure if I do, some number of constraints when you reduce, when you try and get this radial density, and the problem is

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that what it ends up counting is like intersection points of like curves onside like an n-dimensional torus or something, sum something over those points, which is essentially just a constraint,

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a set of constraints, and it's saying that in the large end limit you only get one constraint, which would be cool.

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And then here I guess it's trying to probably show why you get two peaks.

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Maybe this is the minimum, these are the two maxima.

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Of course, two strict local maximas separated by local minimum.

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So I guess that's just the diagram makes that kind of obvious.

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So it's basically saying that you can get the red curve as an integral, and the thing that you're integrating, you can write it explicitly, supposedly.

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I'm surprised that it's got a bump here.

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It's saying that it's doing the marginal of the JPDF, essentially like numerically computing the thing that you've proven analytically.

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It's not really true, I feel.

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I shouldn't have a bump there.

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Again, it cannot explain the new low-order radio station repair.

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These results move to causal questions away from the local singularities and towards global connection coefficients.

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I mean, I am talking to an alien right now, I have to say.

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I mean, I do feel like I'm talking to someone who's done a lot of stuff.

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I feel like if the proof looked like this, I mean, I don't know if you guys saw Terry Tao made a talk.

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And I thought it was a really good talk, actually, because one of the, like, it's not just about proving things.

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It does need to be comprehensible.

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I mean, is it that useful if we get approved of stuff and it's in Clanker language?

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Like, who cares?

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I mean, it's not really that interesting in a way

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because it hasn't actually given you a more interesting underlying structure that shows that that's why it is the way it is.

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But we didn't stop there because, you know, I actually wanted to see even more of what the Clanker was able to do.

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So we gave it an even more difficult problem.

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Not that it solved the previous one.

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And the main goal of this one was to prove whether this two-hump phenomenon stays

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when you take the dimension of the matrix to infinity.

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So as you can see here, it's been thinking for four days.

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Which was a long time.

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I had my computer running 24-7 with a piece of software called...

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What was it called again?

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It's called Amphetamine.

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So it's actually just like a piece of software

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that generalizes some sort of function that you can put in the terminal that like lets your Mac go on forever.

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It's just got slightly better settings to that Mac command.

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But yeah, my computer's been on for a long time, you know, overnight with the fans on.

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And what's come out is quite funny.

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Now, I'm not trying to take the piss, okay?

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So maybe the- the Clankers are probably cleverer than I am, for sure.

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Okay, well I'm not just gonna roll over and like take it like that, but in a lot of ways they are, okay?

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In other ways they're not.

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And you could argue this is one.

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So, after thinking, so it also thought for a long time before this, I mean I'm not gonna like, summarize every single thing that it did, but pretty much it thought for multiple sessions of like,

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loads of hours, and after two days it said, Yes, I've done it.

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I've proven in a rigorous sense that there's at least two distinct modes.

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And its proof is like these inequalities.

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Do you see what I mean?

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I mean, are you for real?

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The completed outward certificate proved that T 1.5 0.4, so these are, I guess,

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some like, locations that you're evaluating something is bigger than 0.00012659100.

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I'm literally talking to a clanker right now.

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Here we've got some more after it thought for a bit longer.

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It gave me some other certified analytic inequalities.

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I mean maybe people do care.

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I mean let me know if you're interested in this.

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I don't really think it's that interesting if you're just you've got two inequalities.

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To me it feels like numerical evidence to be honest.

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Even though I suppose it must be fundamentally different if you're somehow rigorously showing something, it's just like taking brute force to the next level.

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But with this, I had this going on the side, like while I was writing my thesis, I had codex going on this thing, just to see whether, if you just let it go and think forever, whether it gets anything.

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And it thought for multiple times, so this was one session where it thought for four days, I had some other ones where it thought for like three days, and I had one at the beginning that was like one or two days that it thought for,

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so I've been recording this for a long time.

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But yeah, that was what we ended up with.

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We ended up with this.

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So overall it's definitely been interesting seeing how Codex can actually

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work on a project systematically with all of the different research tasks as goals

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and see how it can generate all of these ideas as MD files

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and all of its reasoning and subsequent work so that it can keep this context for future research.

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Of course now

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that I've been given access to both of these models we're going to be putting Opus

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and Fable in on this project as well

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and see how far they can go in trying to understand all of these problems

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because it's a long list of problems all about one thing basically in random matrix theory

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And it'll be cool to see if the Clankers can slowly but surely figure it out.

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That would be very interesting.

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I want to say thank you guys so much for all of the support, like all the channel members and anyone who watches this far.

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But as always, I hope you enjoyed the video and have a good one.

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86 sentences

Vocabulary and speaking notes for this lesson

This C1 speaking lesson is built on the video “We Let $200 GPT-Pro Think for 7 Days on Unsolved Math”. This video has 182 sentences and 2884 words to shadow. The speech runs for 12:17. The speaker talks fast, about 235 words per minute, so expect linked and reduced sounds. 84% of the words are among the 3,000 most common in English; the rest is worth studying before you start.

Key vocabulary in this video

The 15 most advanced words in the video, with pronunciation and meaning:

Word	Pronunciation	Meaning
codex noun	/ˈkoʊdɛks/	An early manuscript book, especially one bound in the modern manner, by joining pages, as opposed to a rolled scroll.
constraint noun	/kənˈstɹeɪnt/	Something that constrains; a restriction.
summarize verb	/ˈsʌməˌɹaɪz/	To prepare a summary of (something).
matrix noun	/ˈmeɪ.tɹɪks/	The cavity or mold in which anything is formed.
numerical adjective	/n(j)uːˈmɛɹ.ɪ.kəl/	Of or pertaining to numbers.
inequality noun	/ˌɪn.ɪˈkwɑ.lɪ.ti/	A condition or state (of social, cultural, or legal matters) that is not equal; especially, such a condition that is thereby also unfair.
bump noun	/bʌmp/	A light blow or jolting collision.
analytic adjective	/ˌænəˈlɪtɪk/	Of, or relating to any form of analysis, or to analytics.
zeta noun	/ˈzeɪtə/	The sixth letter of the modern Greek alphabet (Ζ, ζ) preceded by epsilon (Ε, ε) and followed by eta, (Η, η); or the seventh letter in the ancient Greek…
token noun	/ˈtoʊkən/	Something serving as an expression of something else.
dimension noun	/daɪˈmɛn.ʃən/	A single aspect of a given thing.
codec noun		A device or program capable of performing transformations on a data stream or signal.
coefficient noun	/ˌkoʊ.ɪˈfɪʃənt/	A constant by which an algebraic term is multiplied.
rigorously adverb		In a rigorous manner.
systematically adverb	/ˌsɪs.təˈmæt.ɪ.kli/	In a systematic manner; organizedly or methodically.

Phrasal verbs you will hear

Word	Pronunciation	Meaning
set up verb	/ˌsɛt ˈʌp/	To make ready for use.
come across verb		To change sides; to cross over to work for the opposition.
come out verb	/ˌkʌm ˈaʊt/	To be discovered; to be revealed.
come up with verb		To manage to produce, deliver, or present (something) by inventing, creating, thinking of, or obtaining it.
end up verb		To bring to a conclusion.
go through verb		To travel from one end of something to the other.
reach out verb		To extend one's hand(s) forward; to reach for something.
read through verb		To thoroughly read something to check or proofread it.

Sentences worth repeating

Short, complete lines from the video that you can reuse in everyday conversation:

That's my understanding of it, okay?
I'm surprised that it's got a bump here.
It's not really true, I feel.
Do you see what I mean?
I mean, are you for real?

Grammar in this video

The structures the speaker uses most, with the exact words from the video:

Structure	In the video
Present perfect continuous have/has been + -ing — an action that started earlier and is still going on	that's been thinking · it's been doing · I've been continuing
Present perfect have/has + past participle — a past action that still matters now	has accelerated · have been proven · I've already proven
Passive voice be + past participle — the focus is on what happens, not who does it	being solved · have been proven · have been solved
Relative clauses who / which + clause — extra information about a person or thing	line, which is · subscriptions, which is · minutes, which is

Pronunciation to watch

The speaker uses 75 contractions and reduced forms, such as I'm, I've, we're. Say them the short way, as you hear them.

The “th” sounds: nonetheless /ˌnʌnðəˈlɛs/, logarithmic /ˌlɒɡəˈɹɪðmɪk/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, thesis /ˈθisɪs/
The “sh” and “zh” sounds: dimension /daɪˈmɛn.ʃən/, coefficient /ˌkoʊ.ɪˈfɪʃənt/, experimentation /ɪkˌspɛɹ.ə.mənˈteɪ.ʃən/, gibberish /ˈd͡ʒɪ.bə.ɹɪʃ/, hallucination /həˌluːsɪˈneɪʃən/
Long words — get the stress right: numerical /n(j)uːˈmɛɹ.ɪ.kəl/, inequality /ˌɪn.ɪˈkwɑ.lɪ.ti/, analytic /ˌænəˈlɪtɪk/, coefficient /ˌkoʊ.ɪˈfɪʃənt/, systematically /ˌsɪs.təˈmæt.ɪ.kli/

How to practise with this video

Listen to the whole video once without speaking and note the words you do not know.
Start at 0.75× speed, shadow it sentence by sentence, then go back to normal speed once it feels easy.
Record yourself and compare with the original, paying attention to words like codex, constraint, summarize.

What is the Shadowing Technique?

Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.

Shadowing technique: read the full step-by-step guide →

Shadowing Practice: We Let $200 GPT-Pro Think for 7 Days on Unsolved Math - Learn English Speaking with Video

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