Overview of differential equations [video]
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I've been terrible at math since they started to get "hard" as a teenager, which then compounded into not liking math before and after getting into college for a CS degree.
The degree inevitably requires some math background at the beginning, so I struggled with that a lot, until I discovered 3b1b.
His series on calculus and linear algebra got me through both subjects in less than a week. From 0 to 7 in the scale of 0 to 10 we use in my university, in just 3-4 days per subject. I regret not discovering it before.
What those two series helped me with was understanding why math is important, and how can one solve everyday problems with math, which is something teachers say when you are young but don't actually tell.
Realising that instantaneously motivated me to put in time and effort, which is key when learning math.
The machine learning series also helped me find the answer to the "why am I learning math if I'm just a tech?" question.
On the side, I'm now reading "Computer Networking: a Top-Down Approach" by Kurose, which made me get to the conclusion that we are teaching things wrong. The educational system that brought me here teaches you on a promise of everything making sense once you understand it all, which makes it really hard to understand the pieces in the first place.
The book takes a different approach in that it shows you the final result, and then breaks it down abstraction by abstraction, which makes you eager to know what's going on behind the next abstraction.
If I were to teach a kid how a mechanical watch works, I wouldn't start by the gears, I'd start by the watch itself, and then break it down piece by piece. Once he knows that there's this abstract system that takes some force in one end, and spits some other force in the other, then the kid will want to know what's behind the abstract system, and he'll be ready to know it.
This way I'll keep the attention of the kid till the end, instead of telling him "you'll understand once we finish" after each lesson.
Granted, it doesn't really work as a motivator and those who use it as such are missing a real motivator quite often. Nevertheless, when you think about people, you will find that their actions will make sense once you see all the factors that influence them. Knowing that you probably don't know all factors that affect a person's actions will make you search for the missing parts before judging that person, resulting in a suspension of judgment [1] which is quite valuable actually (not just for human interactions).
And curiosity is the greatest of all drives. I've spent more hours than I can count following curiosity with no other prize but knowledge. Instead, we've come to tell kids that they should memorise this thing for the next week, and move on.
For a given concept, with three layers of abstraction, if we start bottom-up, with the downmost layer, the kid doesn't know what that layer's good for. That happens until we get to the topmost layer. The layers have no purpose until we've finished 4 months from now. They don't make sense.
If you teach kids backwards, once the kid knows the topmost layer, he/she will know what the next layer is good for and in what conditions shall it exist. The _next layer_ has a purpose, which the kid can understand and expect.
The Socratic method works this way. You start with the general purpose of the system, and then you break it down question by question.
(teacher)- What is this?
(student)+ This is a car.
- What is a car good for?
+ It moves people.
- How does it move people?
+ Because it moves these wheels.
- How does it move those wheels?
+ Because... I don't know (takes a car replica), it has these bar connected to those wheels, and this other bar connected to that bar.
- What happens when you spin that second bar?
+ The wheels move.
- So to move the car, you move those wheels, and to move those wheels you spin this bar, but you don't manually spin this bar when you are driving, who does so then?
+ I don't know.
Now the kid is ready to understand what an engine is.
If we did it backwards, and started with combustion, the kid would have no idea what combustion is for until we've reached the car.
i think the fact that you thought a certain way of learning is more appropriate is actually correct - some people learn one way better than another, and that's because in their brain, they model the world in a different way to other people (who also find a different way of learning better). It just so happens that sometimes a person's model happens to suit the sort of teaching method being used in a school, and he ends up looking like a genius.
I can agree people learn very differently about abstract things. Throughout my career I've seen truly remarkable students that had a very different way of thinking about problems that gave them a competitive advantage over others.
https://www.patreon.com/posts/19586800
Discussed here:
I think part of why it can be hard to learn is that one must guess what is in the unknown areas whereas in past problems with limited information (eg in geometry), the puzzle is to derive just enough information on the boundary to solve the problem, so one may not be able to say something about the areas of some figures but might have something to say about their sum.
I found the trick to phase diagrams was realising that they are mostly made of a few interesting features: at every fixed point the diagram is some combination of a circular motion and a scaling in/out motion, or it is some kind of saddle point going out in one direction and in at another (eg the fixed point at the top of the pendulum in the video). Then you just fill in the gaps. This feels hard at first because you can imagine there could be anything in the gaps but it is actually easy because the flow lines do not cross and one cannot have significant turbulence without fixed points (and those have all been found) so only boring things can be drawn in the gaps.
The same is also true with drawing contour diagrams, just replace fixed point with stationary point, and flow line with isosurface.
[1]: https://www.youtube.com/playlist?list=PLDesaqWTN6ESPaHy2QUKV...
Sources for episodes are included: https://github.com/3b1b/manim/tree/master/active_projects/od...
The composer behind for many of the songs is Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue...
3b1b interview: https://www.youtube.com/watch?v=A0RH93XvSyU