i don't know what "should on internet friends" is supposed to mean

means the teachers, author of books etc..

what about them

who are world leading techers.

what does "internet friends" mean?

youtube. Video lectures

The problem is basically "But that good teacher should [do what?] on internet friends." Without telling us something that they are doing, the sentence doesn't really communicate a message.

@123 OK, so you're saying teachers should also give online lectures

but again, for something like math that might just be a waste of time

@JMac Sorry about my bad english. I was answering the previous question. It was supposed to me the verb is there.

a lot of math doesn't lend itself to visuals

and the math that does lend itself to visuals usually requires some software to show, which many teachers might not know

A lot of math just doesn't lend itself to my brain. I'm okay with that

the kind of stuff 3Blue1Brown does for a living

@SirCumference Agreed wherever possible use visuals. Otherwise teach them in a way they wanted to

@SirCumference or it does but only to some people. E.g. the "pictures" of algebraic geometry are famously inscrutable for those not familiar with them :P

@SirCumference Yes 3b1b is an example

@ACuriousMind Are you telling me a layman can't make sense of this:

@ACuriousMind yeah i was gonna mention, abstract algebra probably wouldn't benefit from videos at all

and any cases of algebraic geometry can often just be depicted as pictures in textbooks

animations would be needlessly difficult to create

@JMac bah, at least that's a...shape?

I was thinking of this:

@123 he mainly does geometry and lower level calculus/linear algebra

this is what algebraic geometers call an "intuitive picture"

when you get to the more conceptual ("abstract") parts of math, you often lose the ability to visualize things in that way

@ACuriousMind Literally the first thing on wikipedia for "algebraic geometry" though. Pretty funny to see after you mentioned that.

it's probably the visualization of some fancy variety

but yeah, it doesn't help you understand what it's about at all

i don't see how 3B1B could possibly make a video about an introduction to algebra, e.g. categories or abelian groups, in the same style as his other videos

imo abstract algebra is more understandable until examples are given.

@ACuriousMind What is that supposed to be a picture of anyways lol?

@JMac $\mathrm{Spec}(\mathbb{Z})$

Visuals need to explaintion.

my point is many things in math aren't visual, they're purely conceptual

If one explain with visual it memorized way easily

how would set theory or proof theory benefit from visuals?

@123 math shouldn't be memorized at all, it should be intuited

How can be things understand without creating a pictures in your brain. Your is bound for visuals

@ACuriousMind The picture kinda reminds me of root locus stuff I barely managed to somewhat learn for my system dynamics II class

visuals are not always the correct means to give an intuition

@ACuriousMind It's hard not to feel a bit bad for substitute teachers :P

You just keep making universal assertions that might be true for some people and some topics but you'Re not doing anything to show that are true universally. There isn't the One True Way to transmit knowledge.

you seem to be under the impression everything in math translates to geometry

which it really doesn't

only very nice things do

It's definitely not always better to create pictures in your brain

visuals that our brain can comprehend are too limited for the entire scope of math

But our brain can't understand anything until you make a picture or video in your brain.

you start to notice that problem as soon as you deal with more than 3 dimensions in calculus

We are wired like that

Some structures are very complicated and/or higher than 3 dimensional, and if you spend too long imagining it's structure as something that it isn't you can infer properties about it that it doesn't have

You need to stop asserting how everyone learns or understands things, it's not always true.

You seem to think that we all have identical brains :P

reminds me of that joke about how mathematicians visualize 3d space - they first visualise an n-dimensional space and then let n=3

@ACuriousMind lol i'm keeping that one

@ACuriousMind :O

Not only 3D i am in favor of algebra. Which gives us more than 3D or nD

The numbers in algebra at which you believe are also visuals

basic linear algebra lends itself to geometry nicely

but most vector spaces don't

@123 I'm pretty confident I understand large parts of quantum physics and I literally don't picture anything but formulae when I think about it.

you're not gonna understand infinite dimensional vector spaces via visuals

and those come up everywhere, even QM

The formulae are also visual. We create configuration space more than 3D there is mostly a method at which can be visualized.

in Euclidean space, or maybe even finite dimensional spaces in general, yes

@123 But I didn't need to watch a video to put these formulae in my head!

Formulae are visual insofar as you can picture them in your head, but their mathematical meaning need not be geometric

but how do you visually interpret infinite dimensional spaces?

The numbers we are learnings are also visual without it it is impossible to learn.

Well now I just question your definition of "visual"

:P

@123 what

i've never needed visuals to do addition

@123 There are literally people incapable of mental imagining, it's called aphantasia. You're simply wrong.

without formula how can we calculate anything? formulae are also visual we are bound to visuals by any means

@123 OK, what exactly is your point

videos, visuals, formulae, AI or in person lectures?

because those are all very different things

@SirCumference Calm down :O

I think of things quite visually, and there's even a lot of topics I don't visualize at all.

i'm pretty calm :P

confused admittedly tho

Everything which is possible to deliver maximum information. In my opinion videos are pretty easy to understand

then books

i don't think we're gonna be able to change your mind then :P

i don't say books are not important it as important as it is.

I think a lot of people would argue they learn more efficiently from books than videos.

if you find a way to visually interpret everything in abstract algebra, let me know

@SirCumference I always found difficulty to learn with books. They don't talk may be many other peoples also have similar problems.

I don't really visualise much of what I know about physics

I actually thought I'd end up spending a lot of time trying to visualise when I was preparing to change from chemistry to physics but now I find that often leads you in the wrong direction

@123 i think ya gotta find the right book then

the best ones combine intuition with rigor

@SirCumference Ookay. Sure

Khan Academy's multivariable calc articles, Griffiths EM and Taylor's CM are my top 3, all great at that

I have gone through multivariable calculus mostly understandable to me

@Charlie I picture quite a bit to get an intuition for a lot of things; but I studied mechanical engineering which is basically all macroscopic systems, so it's fairly easy to visualize a lot of things. Some stuff I don't remotely visualize. Like when thinking of stress I don't really visualize it, even though showing stress on 3D objects can be quite visual

Griffiths EM and Tylor CM i definitely read this books.

Oh yeah I certainly did when doing stat mech and or classical mechanics and that kind of thing

Guys sorry about the long discussion in one topic.

But you guys are awesome. You helped me a lot

What is CM

classical mechanics

Ok

I have downloaded both books for reading. You have also suggested for LM. What was that?

@123 Taylor Classical Mechanics

I remember rating my textbooks in undergrad when I had nothing better to do, they'd go like this

Ok from chapter-6 Calculus of variation

@SirCumference lol, "I got so bored I ranked my textbooks" is a special kind of bored

@ACuriousMind the things i will think to do when i procrastinate :P

That is pretty next level

@123 yep

Thanks

An static solution of Einstein Field Equation (EFE), is given. Then I applied the Newman-Janis Algorithm. Is it guaranteed that the rotating will be an solution of EFE?