[dingosity]

2b1b does a video on how you raise e to a matrix (hint: raising something to e is converted into another function where it's not nonsensical to do such a thing.)

In about 10 minutes he explains something my math methods instructor struggled with over two weeks (4 classes). I was resistant to the view that YouTube could host "decent" instructional content for a long time. That video was one of the first that was CLEARLY superior to the instruction I received at a tier 1 research institution.

[gumby]

> CLEARLY superior to the instruction I received at a tier 1 research institution.

It was pretty clear at the R1 institution I attended (MIT) that it is really a huge research lab with a small school attached, and that the educational part, for undergrads at least, was definitely not a priority. The continuing stream of advertising (err, alumni updates) that I’ve received over the subsequent decades don’t seem to contradict this either.

Still glad I went and learned a ton, but it’s not for everyone.

ETA: don’t mean to imply that it might have been good or bad for you, dingosity (how would I know?); just responding in a general way to a singular anecdote/observation in your comment.

[ChuckMcM]

Interesting that we both saw the same thing. I visited MIT as it was on my list of schools I wanted to attend out of HS and realized after that visit that all the "action" there was in research/postgrad so it was a better graduate school candidate than undergrad.

[webshit2]

The undergrads are more involved with research than at most other schools (huge range in quality/difficulty/etc, but the option is there). The teaching quality doesn't really suffer from the focus on research. Blackboard style classes probably aren't much better than any other school, but lab classes were quite good.

[ChuckMcM]

To be clear, I'm not being critical here, it was an impression from my visit. Lots of people I know (like gumby) and others have gone to MIT as undergraduates and really felt it was worthwhile. I really wanted to go to Caltech but only landed on the wait list and settled for USC (who offered better financial aid so there was that)

My kids all ended up at various small private colleges depending on what they found as a fit for their interests and I got to see through them the difference between going to a school with 20,000 students (USC) vs one with 1,500 students (Reed). In hindsight I suspect I would have done better at a more intimate school (harder to have one's slacking off ignored by the administration right?)

[webshit2]

No worries, that's definitely a reasonable impression since there are way more postgrads than undergrads. I just wanted to point out that undergrads are allowed to participate in the 'action'. But of course there is the occasional bad class (Peter Shor is not that good at teaching Shor's algorithm).

[dingosity]

For what it's worth, the lab classes I attended at the public university in Texas I attended after one quarter at MIT was essentially... "here's the key to the lab, knock yourself out. come find me if you have questions." The only reason we didn't make an initiator in the lab that year is we couldn't find the polonium.

[locusofself]

I've been watching some MIT lectures online recently, most of which are about 10 years old. Do they still use those huge a chalkboards?

[webshit2]

Yes usually blackboard or slides. Sometimes iPad + stylus.

[jonahx]

I've had math instruction at the undergraduate and graduate level at top universities, with good instructors that I liked, and I can say without question YouTube is superior. There are many channels with outstanding content. This video is not an outlier.

[impendia]

I am a math professor, and I try to do a good job teaching. Any topic that 3b1b covers, the video will give a better presentation than I would in a lecture hall!

It is part of our job to lecture, but that's not our only job as an instructor, and I wouldn't even say it's the most important one. Our job is to assemble a coherent course -- typically involving lectures, homeworks, exams, etc. where everything is tied together, and where I try to chart a coherent path through the material, so that the students have learned a subject by the end.

Once I was teaching vector calculus, a subject which 3b1b covers, and I showed a couple short clips in class. I also shared additional links with my students and encouraged them to watch. As instructor I try to make good use of any and all helpful resources which are available, and 3b1b is certainly one of them!

[dingosity]

You clearly don't teach at MIT, where the professor's role is to be a glorified test proctor.

[inetknght]

I thoroughly enjoyed watching 3b1b's videos. They're very intuitive for me.

> This video is not an outlier.

I would argue: this video is an outlier in that it's "best of the best". I've seen other videos. While they're usually "good", they're also usually very dry.

[dingosity]

And I've absolutely seen some "meh" videos. Apropos of nothing, I met Jim Blinn a month ago which got me thinking about "the mechanical universe" (from '85-'86) which was one of the first videos I saw that really "nailed it" in terms of using video to teach technical concepts.

So good videos aren't a new thing. But I think there's MUCH more decent and excellent content coming out these days.

[remus]

Personally I don't think it is so clear cut. The value of in-person instruction is in being able to ask questions, and a good instructor will then adapt the content to help answer your questions and give you a rounded understanding of the material. Youtube videos are amazing in several respects (thinking of the reach and production value of 3b1b, for example), but being able to ask questions of a knowledgeable expert is valuable imo.

[beambot]

From my experience at a R1 school, this is only true for late-undergrad and grad classes, not early-stage coursework with massive student-to-teacher ratios. In those courses, you are more likely to get Q&A time with a TA during office hours. So perhaps a YouTube+TA would be better model. Bonus is that you can rewatch lectures again & again until things click.

[chrisweekly]

See also https://betterexplained.com

[SPBS]

Because educational YouTubers only get one shot at bringing their point across (once the video is created, it's there for good) so they really polish every aspect of their video.

Also there's the part where they can't assume their target audience is a bunch of students who have to learn a subject so they really do try to make it as layman as possible.

[veltas]

It's much like open source vs corporate, but not quite. The bazaar really is better than the cathedral.

[analognoise]

Could you share a list of favorites you've collected?

I wish there was a big list that had links to the best lectures and the accompanying book for so many things.

[jhanschoo]

Have a look at 3b1b summer of exposition https://www.youtube.com/watch?v=F3Qixy-r_rQ

[pdpi]

My "ten minutes to understand something I never quite got" moment was his video on Fourier Transforms.

[geokon]

This probably won't be a very popular take.. but to me Fourier Transforms are the perfect example of the opposite problem - where the educators are so hellbound on constructing a visual explanation that they end up brain damaging the students. I've seen this come up several times in my math education.

For the Fourier transform there are primarily two issue:

1. The "complex plane" - this tries to make complex numbers somehow less scary and more intuitive? But it's actually a useless crutch that gives people a false sense of understanding. The core issue is that you can multiply two complex numbers and get another complex number. If I give you two vectors or two points on a map and tell you to multiply them.. you can't - b/c that is not a defined operation. We have no intuition about how to multiple 2D numbers (you might think, oh dot products and cross products! but those are something unfortunately unrelated)

2. "Frequency Space" and an the constant suggestion that your signal is somehow being broken up into it's magical innate hidden frequencies. This is also very deceptive (or doesn't hold up in the discrete case). The basis is selected to be mathematically convenient (ie. orthogonal) and is based on your sampling window and properties of complex numbers. These may correspond to some underlying natural frequency that's occurring - or it may not. But it's not helpful to try to confound the two

If you instead approach the whole problem from a purely mathematical perspective of projection on an orthogonal basis and how to construct complex values that are convenient - then the whole setup is less "fun" but it actually becomes a lot more understandable. You can then move on from there and start asking yourself much more interesting questions about aliasing, ringing, fm/am, phase etc.

I feel it took be ~5 attempts of learning Fourier Analysis to unwire my brain and unlearn these bad visual intuitions that send you down the wrong path

[hpcjoe]

I gotta say I disagree. I got the space change bit quickly (piece-wise linear space into frequency space). Complex plane's aren't a crutch, really they are the firm theoretical basis in which to express the detailed discussion properly.

You could always just switch to the Fourier sine and cosine forms, and avoid all of the other theoretical basis baggage. Sort of like physics for poets (I'm a computational theoretical physicist by training), leaving out the more detailed derivations and background, for a more straightforward approach.

Moreover, the DFT is not the FT. In the limit as your sampled points get very large, it will approach FT. There's a great book covering lots of these things in depth[1], with a pragmatic as well as theoretical approach. I think I gave my daughter this one (math phd student) last year.

FTs aren't merely a change of basis, there is quite a bit more to them than that. For DFT you can look at the process as a sequence of operator applications, but in the FT case this becomes a continuous sequence. Hence the space bits.

[1] https://epubs.siam.org/doi/book/10.1137/1.9781611971514 I highly recommend this book.

[9erdelta]

Can't speak to Fourier specifically -- but heartily agree that a lot of times in math, teachers try to make it "easy" and provide visuals that are actually a hindrance. Instead of understanding _the material_ you understand the _crutch_.

[SkyBelow]

I think there is a special trick that happens many places in math. When you are given a visual, and then some integral or derivative is applied to it, you need to pause and make sure to truly include that in the mental model. Often I find math videos go far too fast through this section and I have to re-watch that specific spot and even pause and just ponder on how does the integral or derivative change what I was working with.

[lahvak]

Ad 1.: Multiplication of complex numbers is a very intuitive operation on complex plane, the moment you realize you need to use polar coordinates.

[marshray]

Link? (please :-)

[suremarc]

I understand Fourier transforms fairly well but his video blew my mind. He has a totally original way of distilling mathematical concepts intuitively.

[hpcjoe]

There's an old joke in math/physics student circles that covers this.

Student goes to see the prof in their office hours to try to understand something the prof said is seen to be trivial. They spend 4 hours working on it, and the student returns to their friends in class. They ask how it went. The student says "yeah, it was trivial."

For those who don't quite get the joke, 4 hours to show something is trivial, tends to not support that the thing is trivial to comprehend.

[pdpi]

To be fair there is a grain of truth to the joke. E.g. Cayley’s theorem is sort of like that — once you get it, it feels like it barely qualifies as a theorem, it’s just a natural consequence of the definitions involved.

[robertlagrant]

Yes. The visualisations and the narration make it so clear.

[eimrine]

Just listen to some music with spectrum visualization, this is a Fourier transform of what you are listening right now.

[jameshart]

That doesn’t provide the full understanding you might think it does. For example: define ‘right now’.

Instantaneously, music is a single sound pressure measurement. That doesn’t have a Fourier transform. It doesn’t have a frequency. It’s just a single sample.

Fourier transforms work on functions. Typically functions in the time domain. And typically (but not always) on that function within a bounded range of time. And the result is another function, this one of frequency.

A spectrum analyzer, though, is showing the Fourier transform of a short snippet of some music. Then a moment later it’s showing you the transform for the next snippet.

Looking at a spectrum analyzer makes you think a Fourier transform is itself a function of time (to some vector of numbers perhaps?). That is not the case. So looking at a spectrum analyzer can give you an incorrect intuition for what Fourier does.

But you can do a Fourier transform on the whole of a piece of music. You’ll pick up frequency components like the overall beat, the bar structure, the verse/chorus alternation.

[beambot]

I think I get what you are trying to say... but the intuition about "this moment in time" is perfectly reasonable for a spectrum analyzer, since it's actually doing a DFT (not continuous from +-infinity) with the last sample (i.e. "now") defining the end of the window.

[jameshart]

The thing that makes a DFT discrete is that it is over individual samples rather than a continuous function - not that it is over a finite domain.

A Fourier transform applied to a brief window of an underlying continuous function is called a ‘short-time Fourier transform’.

And the frequency information a STFT can pick up is bounded on the low end (think, like the opposite of the Nyquist limit) by the length of the window - this is called the ‘Rayleigh frequency’ - if your window is of length t, you can not detect frequencies lower than 1/t. Which is why your ‘instantaneous’ spectrum analyzer (looking at a short burst of maybe 0.05s of samples) for your 120bpm EDM doesn’t pick up a frequency component at 2Hz - even though that component is there in a Fourier analysis of the whole piece. It can only measure down to 20Hz. Which is fine because that’s also roughly the limit of the part of the song ‘function’ that we hear as ‘tone’ rather than ‘rhythm’.

[beambot]

Respectfully, your characterization of a DFT's infinite domain conflicts with the definition of the DFT -- it is defined as a finite sequence, and that's how it's used in common industry usage. Case in point:

https://en.wikipedia.org/wiki/Discrete_Fourier_transform

> In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples [...]

Related: Showing energy content (i.e. DFT) versus time -- aka spectrograms: https://en.wikipedia.org/wiki/Spectrogram

[campground]

That’s a good place to start building intuition, but it can also distract you actual understanding. What that visualization really is is a bunch of arbitrarily bounded Fourier transforms of little windowed slices of a piece of music. The true Fourier transform of a time bounded piece of music is a single unbounded function over an infinite frequency spectrum.

[pdpi]

Well, yes. That part is obvious enough. The interesting part is how you build the mapping from time domain to frequency domain. That is the part that never clicked for me.

[eimrine]

Maybe that's because a sinusoid of any frequence has its "roundness" and that is a matter of "hunting" which that formula performs every infinite-small moment to an x/y graph of your favourite music composition.

[Sohcahtoa82]

That only tells you what the result of a Fourier transform is. It doesn't tell you how it works.

[rjh29]

> I was resistant to the view that YouTube could host "decent" instructional content for a long time.

There is SO much good content on there though. I can only guess "the algorithm" was hiding it from you. Sadly we often have to rely on sites like reddit or HN to find these creators.

[VBprogrammer]

While we're at it. Ben Eater is the equivalent for Computer / Electronics fundamentals. Check him out of you haven't before.

[aeternum]

the algorithm is actually quite good but you need to give it signal. Most people prefer entertaining videos over educational videos. If that's what you tend to click, that's what it will show you.

Entertaining is hard to resist, so just create a second account and curate the history of one of them.

[rjh29]

It's good for me too, but it took a very long time to train it to stop giving me 'YouTube face' videos, clickbait bullshit, Linus Tech Tips, politics, Amber Heard etc. And it still sometimes serves that up if there's a new topic.

I also find my feed doesn't change that much, so the downside of being picky is that you might miss out on some good content.

[codetrotter]

> a video on how you raise e to a matrix

Video link https://youtu.be/O85OWBJ2ayo

[gowld]

> hint: raising something to e

raising e to something

> is converted into another function where it's not nonsensical to do such a thing.)

That function is the power series expansion, taught in your calculus class. https://en.wikipedia.org/wiki/Matrix_exponential

> CLEARLY superior to the instruction I received at a tier 1 research institution.

3B1B is a tier 1 educational institution.

But also, videos often feel better than classes, because videos don't have homework, so you don't actually have your feeling of understanding tested.

[GuB-42]

What 2b1b, as well as other science YouTubers do is that they give you insight that more conventional teaching may lack. What YouTubers don't offer however is training, as in, the sometimes boring part where you do exercises.

They are complementary, an entertaining YouTube video may first give you motivation to get into a seemingly boring subject. It may also help you after you took your classes, to help make elements fall into place, but it only works if you have learned the elements beforehand.

As educative these videos are, they are entertainment, and they are driven by the rules of entertainment, they wouldn't get much views otherwise. So it leans heavily on accessible and clever explanations, surprising results and relatively short form, the boring stuff is often left out, and unfortunately, some of it is necessary to progress. Most science YouTubers admit it, and urge everyone to take school seriously.

I have never learned a field just from YouTube videos. It either left me with superficial knowledge (which is better than nothing) or acted as a way to consolidate knowledge I already had.

[jackphilson]

perhaps we should remodel our educational system towards content that had a great deal of time spent creating it rather than a slideshow created last night. if only there was this medium that allowed information to be duplicated all over the world instantly so that every professor doesn't need to create their own content

[jerf]

I'm actually some combination of flabberghasted and annoyed that we're in 2023 and computers have still not been terribly successfully deployed into any educational setting.

I suppose the idea that we should look to the results rather than some Ivory Tower education professor's ideas about what constitutes good teaching is just too foreign for the establishment as a whole. I've got armies of PhDs A-B testing whether they can extract .001 cent more per ad from me, but it's too much to ask of society as a whole to check into the question of what curriculum works better. Grant shouldn't have to be all but forcing his way uphill into this space.

[frankfrank13]

Much the same experience for me, although sans tier-1 lol. He's clearly gifted, for a long time I think wrongly assumed everyone who makes content is not.

[Juicyy]

true for most of calc and a lot of electrical. Institutions failed to discuss why you need to do things. so much of it is here is Taylor Series or integration by parts. but nobody really knew what the application for this stuff was. just follow some rules and plug in some numbers.