My advice for math is that it’s often possible to think you understand something even if you don’t, so it’s good to do at least some exercises. Also, the methodology, and general “mathematical maturity” is often what you’ll reuse the most in research—being able to reason by following specific allowed/disallowed steps, and knowing that you can understand a claim by reading Wolfram Mathworld, Wikipedia, textbooks, etc. So to some extent it doesn’t matter so much what you learn, as that you learn something well. Having said that, the first half of a math textbook tends to be much more useful than the second half—there are diminishing returns in each subfield.
For programming, the same is often true—what you’re aiming to get is a general sort of maturity, and comfort with debugging and building programs. So probably you want to mostly read tutorials for initial few weeks, then mostly do a project after that.
In both cases, I agree that a tutor is super-useful for getting unstuck, if you have the luxury of being able to afford one.
My advice for math is that it’s often possible to think you understand something even if you don’t, so it’s good to do at least some exercises. Also, the methodology, and general “mathematical maturity” is often what you’ll reuse the most in research—being able to reason by following specific allowed/disallowed steps, and knowing that you can understand a claim by reading Wolfram Mathworld, Wikipedia, textbooks, etc. So to some extent it doesn’t matter so much what you learn, as that you learn something well. Having said that, the first half of a math textbook tends to be much more useful than the second half—there are diminishing returns in each subfield.
For programming, the same is often true—what you’re aiming to get is a general sort of maturity, and comfort with debugging and building programs. So probably you want to mostly read tutorials for initial few weeks, then mostly do a project after that.
In both cases, I agree that a tutor is super-useful for getting unstuck, if you have the luxury of being able to afford one.