[throwlaplace]

>As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math

Me too so now what? I don't think your credentials give you any real authority but just make you look like you're gatekeeping.

>Doing the exercises is playing with the objects to try to answer specific questions.

Great so then we're in agreement: playing with the object is doing the exercise.

The funny thing is that at one time I actually did all of the exercises in volume 1 of apóstol's calculus. You know what effect on me it had? I was so bored I didn't read volume 2. And today I'd still need to look up the trig substitutions to do a vexing integral.

[ColinWright]

> I don't think your credentials give you any real authority ...

It wasn't intended to, it was to provide a context for my opinion.

So let me state my opinion as clearly as I can, and then I'll leave it.

* Math is a "contact sport" ... you have to engage with it;

* Reading books is not, of itself, engaging with the math;

* Watching math videos is not, of itself, engaging with the math;

* Well designed exercises are a valuable resource;

* If you can easily do an exercise, skip ahead;

* If you can't do an exercise, persist (for a time);

* Ignoring the exercises is ignoring a resource;

* For the vast majority of people, doing the exercises is an efficient way to engage with the material;

* To say "ignore the exercises" is, for the vast majority of people, an invitation to not bother engaging with the subject;

* Doing all the exercises is probably a waste. Doing none of them is an invitation to end up with a superficial overview of the subject, and no real understanding.

[daxfohl]

See? It's pretty hard. This is what I've been dealing with for the last 20 years of on and off trying to get through the bigger Rudin book and a couple others.

Just reading doesn't get much at all. Not even a superficial overview. I tried it. It's essentially a meaningless combination of words after a certain point.

Reading extremely thoroughly is actually marginally useful. Stopping to think, do all these assumptions matter, why, what if one of them changes, etc, pencil in hand, making notes, testing things out. I've managed to "understand" the topics when doing this, and so far it's been the highest ROI method. But it does still leave one feeling like something is missing. Just because you can sight read music doesn't mean you're an expert on the piano.

Doing exercises is a huge jump on investment, and the return on that investment is a bit questionable from my experience. A couple reasons: first you don't know if you did them right. If you did them wrong then that's negative ROI. Second you don't know what a "reasonable" workload is. It varies by author. Is it three problems per chapter, is it all of them, are some orders of magnitude more difficult than others? Without some guidance it's hard to know if your difficulties are due to not understanding basic material, or due to that problem being a challenge geared toward Putnam medalists. So they may cause you to question your understanding and thus mentally roadblock you unnecessarily. And finally with proofs (and this may be a me thing), it's pretty easy to say "I guess this is okay(?)" and move on, even if you're not sure. Since nobody is ever going to review it, and it's just a homework problem, it's very very hard to will oneself to make sure every assumption is correct and you're not missing anything, even if you feel like there's a good chance you are. Or perhaps I just don't have the constitution to do so.

So while I think doing exercises is necessary for a deeper understanding, I don't know whether the ROI is worth it outside of a classroom perspective. You need feedback for exercises to be beneficial. At least, I feel like I do.

Finally, is even taking a class that useful if the end state is that two years from then you'll have forgotten most of it and so what was the point. Can you claim knowledge of a subject that you've never actually used beyond some homework problems and exam questions, or is this still a superficial understanding? Having an ends where that knowledge gets used seems critical.

I feel like I have some knowledge but I don't feel like I'm there yet. But I don't know if I know where there is. Maybe that's the biggest challenge. Does completing a Ph.D. even get you to there? No idea. But, I guess it's up to the individual to decide what they want out of it. Nobody can determine that for you.