> After scratching the itch of knowing that something's been written completely rigorously, it's easy to call a work done when maybe we shouldn't call it done yet. And there's contexts where this is maybe OK, research comes to mind...
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
unpopular opinion: imho there is no substitute for deep learning and doing problems. Math is unlike other subjects, where you have to actually do it head on; you cannot just passively consume it like listening to a history podcast or watching a video.
The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do. Learning by watching is many orders more popular than learning by doing. Probably because it makes you feel smart without having to put that knowledge to the test, so it seems like a better way to master new concepts.
There are many folks on HN who watched a YouTube video or read a Quanta article about some mathematical concept and run with that.
Personnally, I watch 3Blue1Brown's videos because I find them very enlightening. They're are also very nice to watch and listen to, the production quality is very high. I don't pretend learning a lot of maths by simply watching them but I like watching them anyway for what they are: videos about maths problems and maths concepts.
> The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do.
I don't think this is true.
First of all, the fact that watching videos is not a sufficient way to learn math does not mean that videos cannot be an excellent supplement for learning math. 3B1B's linear algebra videos for example are wonderful tools for gaining geometric understanding and motivation behind why things are the way the are in linear algebra, but of course they won't teach you how to invert a matrix. (He even says something to this effect explicitly many times throughout the series, if I recall.) Not everyone who watches those videos is engaging shallowly with math just because they're watching videos about it, there's plenty of value for serious students as well.
Your next argument might be that the sheer number of views indicates an unrealistic volume of serious students. This is true, but it still doesn't indicate that people think watching videos is a good way to learn math; in particular, nothing says they're watching those videos because they're trying to learn math. They're beautiful and interesting and well put together videos, and I'm sure many people enjoy watching them passively fully conscious of the fact that they aren't actually deeply learning math. I don't think there's anything wrong with that!
I'm sure there is a subset of viewers who think they're actually learning math by merely watching videos, but I'm willing to bet it's a relatively small fraction.
I think a lot of people, sadly, imply that they think this, even if they don't explicitly say it.
I recently tried to obliquely bring this up to a friend and he asked out loud, "Sometimes I do wonder if it's better to watch YouTube videos or do problems. I'm not sure."
This has been one piece of acedata representing an aggravating trend in my personal life. It could just be my attracting these types (which I've been fretting about since I'm thinking wtf is wrong with me then), but other examples are family members even expressing this bad habit with something as dumb as Marvel. They read forums about comics without ever reading them and feel possessive enough about the genre to get pissed when I, who...does read comics..., criticize the Disney Marvel movies.
I'm hoping it's just me over indexing for it since it bothers me, but yeah - I have found that there are certainly people who just consume content without practicing or implementing the thing they're supposedly learning about, and then think they've actually learned.
ofc ymmv - it could just be my strange social circles.
I heard the following nice metaphor from a professor: learning mathematics is like learning to ride a bike, at some point you just need to get on the bike, you don't just study some "theory of bicycles".
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
Who thinks otherwise?
There are many folks on HN who watched a YouTube video or read a Quanta article about some mathematical concept and run with that.
I don't think this is true.
First of all, the fact that watching videos is not a sufficient way to learn math does not mean that videos cannot be an excellent supplement for learning math. 3B1B's linear algebra videos for example are wonderful tools for gaining geometric understanding and motivation behind why things are the way the are in linear algebra, but of course they won't teach you how to invert a matrix. (He even says something to this effect explicitly many times throughout the series, if I recall.) Not everyone who watches those videos is engaging shallowly with math just because they're watching videos about it, there's plenty of value for serious students as well.
Your next argument might be that the sheer number of views indicates an unrealistic volume of serious students. This is true, but it still doesn't indicate that people think watching videos is a good way to learn math; in particular, nothing says they're watching those videos because they're trying to learn math. They're beautiful and interesting and well put together videos, and I'm sure many people enjoy watching them passively fully conscious of the fact that they aren't actually deeply learning math. I don't think there's anything wrong with that!
I'm sure there is a subset of viewers who think they're actually learning math by merely watching videos, but I'm willing to bet it's a relatively small fraction.
I recently tried to obliquely bring this up to a friend and he asked out loud, "Sometimes I do wonder if it's better to watch YouTube videos or do problems. I'm not sure."
This has been one piece of acedata representing an aggravating trend in my personal life. It could just be my attracting these types (which I've been fretting about since I'm thinking wtf is wrong with me then), but other examples are family members even expressing this bad habit with something as dumb as Marvel. They read forums about comics without ever reading them and feel possessive enough about the genre to get pissed when I, who...does read comics..., criticize the Disney Marvel movies.
I'm hoping it's just me over indexing for it since it bothers me, but yeah - I have found that there are certainly people who just consume content without practicing or implementing the thing they're supposedly learning about, and then think they've actually learned.
ofc ymmv - it could just be my strange social circles.
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
-The goat paul halmos
You are probably correct.
https://en.wikipedia.org/wiki/Deep_learning