[crdrost]

Yes. Well, examples. Good homework assignments give you some examples to sit in your head, not all homework assignments are good though.

The way I like to tell people this is, you know that thing where you want to take your kids or your students and make sure that they don't have to go through the same pain that you did to learn what you had learned? That feeling?

If you have ever had that feeling you must understand that it was horribly mistaken. The problem is precisely that learning and pain go hand-in-hand. We learn abstractions precisely because they relieve a sort of confusion, a sort of difficulty, they organize the pain that we have experienced and make it tidy and less painful. Even as kids abstractions like “this is what it means for the stove to be on” work this way, not that you have to get burned, maybe you just need to be yelled at, but it organizes that pain of being yelled at and tells you when are you getting yelled at and why.

Not all learning is this way, for example memorization and repetition... but to a first approximation you have to get lost in the forest before the landmarks on the map are recognizable.

Corollary: you CAN tell people things, but it might be more involved than just a casual conversation. Either you have to establish a shared context, tap into the pain/confusion that they already have... Or you have to get them interested enough that they will follow you down a rabbit hole of confusion and difficulty so that you can finally explain the thing.

[moring]

> The way I like to tell people this is, you know that thing where you want to take your kids or your students and make sure that they don't have to go through the same pain that you did to learn what you had learned? That feeling?

I am not yet convinced. Let me take the following example because I remember it well: I tried to learn about geometric optics and lenses. To do so, I downloaded multiple lecture PDFs from the internet and read them.

All of them started to explain lenses by describing how large the image of a real object is, defining focal length and such. Literally not a single one of them even defined what an image is. The whole talk about image size and focal length and magnification and what not was totally worthless because it was all based on the same fundamental word that had no meaning. In the end, I tried to make up my own definition that was consistent with all those PDFs, but it left me unsure if I got it right.

The "pain" was there, but there was no relief from it, and I am still convinced that that simple definition could have avoided it. If I were to teach optics, I would give such a definition, and I am still convinced that it would help with the pain.

The other example was one of those PDFs that showed a real object that was "wide" along the distance axis from the lens, and so by my understanding should have an image whose magnification changes along the distance axis, but the explanatory picture in the PDF showed equal magnification everywhere. Today I am convinced that this "explanatory picture" was simply wrong.

Again, there was "pain", no relief (because I could not be sure), and I am still convinced that a learner can be saved a lot of pain when you just exlucde factually wrong content from the learning material.

[melagonster]

this why Socratic questioning is good when you have a personal teacher.

[jrumbut]

I strongly agree with your main point that sometimes valuable learning is painful. I've found a lot of mathematics is painful beyond "this is frustrating and difficult," but in a more profound way.

I would say though, about abstractions, I'm not sure hot stove is an abstraction. I think hot is an abstraction that allows you to apply the lesson you learned on the stove to the toaster and the coffee maker and all the other hot appliances you encounter.

[maCDzP]

If you wouldn’t mind, could you elaborate your point about mathematical being painful?

[jrumbut]

I don't have a great way to put it in words, but the intuition is that you are putting enormous mental energy into something that doesn't give anything back. Being able to solve heat equations doesn't make you warm.

Other subjects are the same, obviously, but when writing a history essay you can BS and not think too hard and skate by with a C. With a math problem, if it's difficult for you, you need to really put your whole self into solving it to make any progress at all. And I think that can be painful, in a certain way.

[andai]

I was studying calculus not too long ago and arrived at the conclusion that drawing is actually way harder than math. In math, you learn how something works, and then you can do it. In drawing, you can literally see the "solution" right in front of you (e.g. your own hand that you are drawing) and still get it wrong for years (and even then, only asymptotically approach perfection).

That discrepancy between the goal (photorealism, or at least beauty) and the result (ugly art) is very painful for me, to such a degree that it discouraged me from putting much effort into art, despite my love for it.

[jrumbut]

Drawing is similar (and a lifelong source of frustration for me). I can see the same experience playing out in different disciplines.

I would say that in the kind of math you study after calculus it becomes more common to have that "I know what the solution should look like but I can't seem to get there" feeling.

[wizzwizz4]

Don't try for photorealism straight away. Pick a more abstract style, and work on that: change things around until you find it fun. When you learn to follow your whims, that's when you'll start getting better (perhaps even quite rapidly).

This isn't to say give up on photorealistic drawing: but don't bludgeon yourself with it. If you don't find it fun, you're probably not at a point where practice will make perfect.