I think setting attainable goals and working towards it helps. It is very easy to lose motivation if you don't know what you are going for..
I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
Nah, it's totally possible for newbies to pick high-level long-term goals.
This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them.
I mean, you can disagree all you want, but this is in fact how many people learn things.
Of course this is in the context of choosing research problems to strive towards in math. If you tasked yourself with solving an open problem in math, it's more likely than not that, without any collaboration, you'd have no idea how to even work towards the goal due to all the unknown unknowns. If your goal is something concrete that can be augmented with mathematics, then yes I agree that goal setting can be useful. It doesn't take a volume of missing domain knowledge to develop that kind of goal.
People just assume learning math is the same as learning everything else. That is not even remotely true.
> everything else that’s worth learning: hard!
I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.