Lessons by 3blue1brown
3blue1brown.com
3blue1brown.com
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.
Video link https://youtu.be/O85OWBJ2ayo
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.
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’.
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
A spectrogram remains a visualization of a short time Fourier transform at a number of points in time. In practice usually produced using a DFT because discrete samples are what you have to work with.
https://en.wikipedia.org/wiki/Discrete-time_Fourier_transfor...
Pedantics aside: Spectrum analyzers are computing DFTs over a finite window, and it's perfectly reasonable to think of these as (an approximation of) power spectral density changing over time.
The thing I am railing against here is the idea that you can just look at a spectrogram to grasp Fourier. You can’t. It is an advanced application of Fourier transforms that creates a visualization of power spectral density over time but it is not a (simple) Fourier transform of the underlying data.
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
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.
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.
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.
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?)
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.
Entertaining is hard to resist, so just create a second account and curate the history of one of them.
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.
> 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.
So good videos aren't a new thing. But I think there's MUCH more decent and excellent content coming out these days.
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!
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.
I wish there was a big list that had links to the best lectures and the accompanying book for so many things.
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.
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.
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.
I watch a lot of educational YouTube videos. But I've come to see them as more entertainment than education. Not in terms of the quality of the videos (which is often excellent), but in terms of what I retain. Sometimes I come across YouTube videos that YouTube claims I have watched, and I don't remember the first thing about them.
I think it's difficult to learn while being just a passive listener. You have to get your hands dirty playing around with the concepts. It would be so cool if these high-quality "lecture" videos were accompanied with high quality exercises.
One thing I do is try to connect the concepts into textbooks I do have. Say I have an elementary proof about a cross product: try and tie that into the stuff about wedge products and exterior calculus I just learned about. Line integrals? Perfect opportunity to practice working with one-forms.
Another option for the mathematically inclined is to pause and prove everything if you feel you aren’t getting it good enough, or to build up your own visual picture from first principles as much as possible rather than memorizing the video’s presentation.
Although I find instructional videos helpful in understanding concepts, the knowledge is fleeting. Exercises would be helpful, but IMHO, sometimes you need the support of a cohort or TA to solve problems in a different way.
MOOCs are pretty good. They have instructional videos, exercises and a cohort. Still, they seem to be missing something. AI will push online learning to 2.0.
I’m convinced that he would welcome and likely support any effort. I’m assuming he has something in mind or underworks, either building something himself or working with Brillant (the de-facto leader in active learning and a sponsor for all science education channels, including 3b1b).
There was an effort in that direction with last years’ _Summer of math_, where he asked students to make their own video.
Sabine Hossenfelder (physicist, of Science News fame) has made content for Brillant, branded under her name, so it could be an option to host it there. I suspect Grant having a website is a precursor to a homegrown option.
Either way, I’m assuming he has all the encouragement and the unlimited support of anyone here —— he certainly has mine.
I don't blame them. Educators are in a perpetual cat-and-mouse game with their students to avoid cheating. However, in an era with Chat-GPT (and even before), cheaters can always cheat if they want to cheat. If a teacher decides to teach calculus using 3B1B's videos as reference, then they should make associated assignments available for other educators to use.
There is no field better suited for opensource than education, yet tribal knowledge remains either inaccessible or under-utilized. It's a shame. And no, "online courses" are not a substitute for real coursework. Watered-down wish-fulfillment QTEs is all they are most of the time. There are some excellent online courses (MIT OCW, NPTEL), but they're the exception. And even then, online courses have yet to replicate the essential support of a human educator or peers in learning.
I look forward to seeing schools that fully construct a curriculum around curating the best internet resources, with the teacher being there more so 'conduct / direct' rather than getting their hands dirty figuring out how to optimally teach the same thing every teacher has taught for the last 100 years. It has the advantage of reducing teacher overhead and improving the quality of teaching. win-win.
All that being said, I've found that 3b1b videos tend to bound the intiution you can get on a topic. Most of his videos tend to reframe an idea in some sort of interesting way that can be easier to grasp without rigirous definitions and theorems. If your goal is to just understand the ground level of the idea, then this intutition is really great.
I've found that this "visual" intuition becomes unhelpful as soon as you try to grapple with a more advanced version of the topic. Your understanding of the topic is often completely limited to the way he frames it, and becomes quite useless when you leave that limited framing. He doesn't teach you "fourier transform", he teaches you a very specific context of the fourier transform which doesn't generalize well.
Again, this is still super valuable if you just want a broad understanding, but not particularly helpful for a comprehensive education of a topic.
Can you give a specific example?
Completely limited?
My framing of human learning goes roughly like this:
1. When we are motivated to solve a particular problem, we can draw upon a multitude of “frames” (one example being metaphor, another being an equation, another being a visualization) to help us.
2. People sometimes cannot shift frames of reference effectively (i.e. perhaps fixating on only an few approaches). This is not the fault of the particular frame(s); this is a limitation in how people use them.
3. To our brains, almost everything is a “model” … a way of conceptualizing, deciding what to pay attention, what to ignore, what questions to ask, and when to recognize we aren’t getting anywhere.
4. Some models / metaphors are better than others for certain people and contexts.
- Some are better connected to other models, allowing more fluid thinking.
- Some are more rigorous.
- Some are simpler.
It depends on so many things.
I don't think the limited understanding is useless once you leave the simplified framing. As an example, linear algebra is far more than just 2/3D continuous space that he covers in "Essence of Linear Algebra", but I think having the solid visual intuition for those simple cases gives you a foundation to learn the less visualizable versions of linear algebra.
You might be one of those people that can do math purely symbolically, by parsing and manipulating expressions, a bit like regular language and you don't need your brain to build a mental image for the things you work with.
I am intuition-first kind of person, and let me tell you, intuition is not a crutch, it can do things where symbol manipulations would take orders of magnitude more effort.
It is actually possible to imagine and manipulate highly-complex, multi-dimensional things that don't have equivalents in nature.
Imagination is a muscle used a lot when thinking intuitively, some of us have it developed to a ridiculous degree.
I will say though that not every topic can have a sensible visual model to it, and being able to acquire symbolic intuition is probably essentiall to go far in any field.
1. You don't understand a concept 2. You go to your professor's office hours with a question and ask them to explain. 3. They draw some squiggles on the board and walk you through the issue 4. You say "Aha! I get it! Thank you for explaining!" 5. Two steps out the door you've completely lost the idea and need help again.
You haven't internalized the idea, you're still just attached to someone else's idea. To understand something fully, you have to really take it in and play with it; find the holes and the limitations of the model.
Teaching is certainly a lot different today. Decreased attention spans, underfunded schools, and ChatGPT make it a lot harder, but online modules and tutorials like these make it a lot easier.
What's the use of arcane knowledge about prime numbers if you are not mathematician, and even then, one should question its importance?
Time is often better spent learning about crucial factors of success such as goal-setting and self-discipline. Additionally I would say that learning about social skills like human interaction, power dynamics, and body language is a more productive use of time.
And what about focusing on the tasks that you really need to or want to do?
Note: I often watch those kinds of videos, but I feel bad about it because I know it's procrastination. In my daily life, I almost never use this kind of knowledge.
Because learning for its own sake can be intrinsically rewarding, and even fun?
> I often watch those kinds of videos, but I feel bad about it because I know it's procrastination.
Don't feel bad about it. Give yourself permission to enjoy just learning stuff that you find interesting.
> Learning and understanding abstract concepts won't make you more successful, ...
As someone else has already pointed out, not everything has to be a hustle.
> ... increase your SMV and give you access to better mating opportunities.
OK. Not everything needs to be seen through that lens either. If that's important to you, sure, work on it a bunch. But surely it doesn't have to be only thing you ever work on?
Plus, you never know - knowing about a few areas of arcane stuff you like to geek out about might end up making you a more interesting person anyway.
The basketball player (Michael Jeffrey Jordan) or machine learning professor (Michael Irwin Jordan at Cal Berkeley)?
If the former, surely we have better role models for teaching than a basketball player. Apologies for the prodding, but cross-domain idolatry gets me riled up.
For me and thousands of others, Grant Sanderson, creator of 3b1b, is the archetype of an internet-era world class educator. Thanks Grant.
> … and youtube/the internet will allow us to find those people and reward them appropriately
Yes, there are many others across various fields. While profits can be a useful motivator, we don’t want that to be the only mechanism. Education isn’t only about the most profitable topics or the students already best situated to learn.
I learned most of the topics in those videos well enough to pass highschool/university exams, but very often the real "aha!" moment came after seeing lines transform from one into another, planes deforming, or boxes being subdivided in 3blue1brown video.
1) When I learn something, I like daydreaming about it and coming up with a mental model of how it is in my own head. So, having someone else's vision and animations used can screw up my own mental model and make it harder for me to forge connections between concepts.
2) I enjoy rigorous math more. I find it hard to make logical leaps and assume properties about things unless they were previously established as facts. This is why I like analysis a lot, althought I'm not great at it. I think 3B1B is more informal when explaining things, and the lack of rigour confuses me over time. Especially in a longer video.
Be aware that many people don't watch videos to learn, they watch to be entertained. They think that they are learning, but really they are just watching animations. In the end, they can't actually do any of the math. They haven't actually learned anything. In the same vein, that's why TED talks are so popular. It's like eating dessert... all sugar and no nutrients.
I'm not saying that 3b1b videos aren't instructional, but rather that they are so well put together that they can satisfy the entertainment-seekers while still offering meaningful content.
The question is, then, why don't you get anything out of the videos? That brings us back to the problem of mental models of understanding.
If someone teaches a subject from a completely different perspective, it may be that it confuses you as your brain tries to harmonize the two conflicting mental models, and the video moves along too quickly for you to be able to benefit from the realignment.
This is not an attack on you, btw. In fact, I believe that this realization helped me a lot as an educator to be patient when teaching. Furthermore, it does not mean that your initial understanding is wrong, but that it is a different perspective than that which is being presented by the instructor.
Alternatively, it may be that the videos move through a particular concept too quickly. I find this is an issue, too. Once I understand something new, I like to almost daydream about the concept, exploring it mentally so that I understand it from a lot of different angles. I think of it in different situations and to see if my intuitions make sense, and if they don't I ask for clarifications. You can't do that when watching a video because it doesn't stop to answer your questions.
I like 3b1b videos. Some of them I can binge on all day simply because they "click" and make sense. Others I have had to spend days (or weeks) thinking about in order to digest and understand the perspective that Grant is trying to convey. They are definitely high value and I greatly appreciate Grant's work, but I definitely have to work, too, in order to benefit from some of them.
In my evaluation of him I mentioned this, but added that he was probably wasted on 3rd and 4th year math majors, and should be teaching more intro classes where the quality of instruction was markedly worse.
Back when I used to teach math classes as a grad student, I realized that a teacher has to actively work to keep that "mental catalog" for students. The problem is that longer you are in the math world, the clearer the subject becomes. The result is that to you the teacher it really just becomes braindead obvious what's going on. As time goes on, you have to fight this "problem" more and more to keep touch with the issues your students are having.
When my high schooler come to me with math problems, the first thing I do is look at the failures and figure out what higher level step or concept is missing in the brain. Then I explain that, sometimes more than once with different approaches. Once the higher level thing clicks, I can just sit back and mostly watch.
Sometimes I'm not sure how a concept clicked, but something did and the right answers start coming out.
- un-abstracting abstract concepts by creating concrete examples, which end up taking the focus away from the abstract version
- serial timeline, that is like a blackboard being constanty erased instead of adding to one big construction
I think it’s absolutely the case that some lessons work for some people at some times for some topics and don’t for others - but that’s not likely down to any innate ‘learning style’ that one person has over another. The same approach on a different topic on a different day might have the exact opposite effect in terms of who gets most out of the lesson.
I think 3b1b videos constantly break this rule; there is a LOT of going on visually, while the narrator is talking non-stop about some loosely related things or trivia. More often than not I have to re-watch segments multiple times because I cannot understand what the hell am I supposed to be paying attention to.
I wish we put more resources into educational materials of this level (and with this kind of universal access) as a country.
I'm not sure that's possible. He is free to create videos on his own time, when he has a compelling idea how to present a topic. "put more resources" on a project sounds like funding with a specific agenda and plan - a committee-determined set of topics to cover and then push on creators with a timeline. You can certainly get good results doing that with the right approach and people, but I don't think it will turn out as good as this. For example, 3B1B gut used to work at Khan academy. Not saying Khan is bad, just that 3B1B is better if less prolific.