tgt said:
What general advice would you give to a keen young (upper high school and undergrads) student of mathematics?
Study mathematics. Most high school curricula include almost none, just Euclidean geometry. Mathematics is proofs. The usual high school algebra, trigonometry, and calculus classes avoid proofs, teaching instead just lots of rules and techniques. Completing these classes, even with excellent grades, tells you almost nothing about whether you like or are good at actual math.
I would say most important is to develop your mathematical intuition by doing and reading as many examples and problems as possible.
Depends on the problems. I would say that problems that take at least fifteen minutes of hard thinking are worth doing, better if they take longer. The improvement of your mathematical intuition comes from thinking about how to solve such problems, especially the wrong paths you take. This is why people who give up and look at answers too soon don't learn as much. If you are susceptible to that temptation, don't use materials that have answer keys.
Also, recognize that generic advice may not apply to you. Some people love pictures and examples, which others find them irrelevant and unenlightening. Some people find proofs uninteresting -- these people are destined to be something other than mathematicians. It's best to understand yourself as well as and as soon as possible.
From my experience, even if you were very pedantic with reading and understanding proofs, without a solid "mathematical background" and mathematical intuition, it doesn't leave a large imprint in your brain. To me the mathematical background and intuition is gotten from doing lots and lots of problems starting at a basic level.
Not to put too fine a point on it, if you have no understanding of proofs then you have no "mathematical background". You develop one by understanding proofs, one tool for which is working difficult problems.
For an extreme example, it is not good to get someone to start learning set theory before they have a good grounding of algebra, analysis etc.
On the contrary, learning elementary set theory and logic and other foundational topics is an excellent way to start.
You might find it useful to read
Measurement, a fairly recent book by Paul Lockhart.
But what are some over arching goals that students should aim for (apart from the obvious of aiming for the highest marks).
This too is quite wrong. Aiming for the highest marks is a useless waste of time. You should aim for understanding the subject you are studying, and good enough marks will come from that. There are many ways to go about getting high marks, and most of them are of no use other than for getting high marks. Such knowledge is of little value once you are done with being a student. The understanding of the subject should be your main goal, as its utility is long term.