Advice is what we ask for when we already know the answer but wish we didn’t. (Erica Jong)
Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages).
Disclaimer: The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommend discussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice. Also, it should be clear that most of this advice is targeted towards academic careers in mathematics; of course, there are many other career options available besides this, but I have no particularly informed advice to offer for such alternatives.
• Primary school level
• High school level
• Undergraduate level
• How can one become better at solving mathematical problems? Note that there is more to maths than grades and exams and methods; there is also more to maths than rigour and proofs. It is also important to value partial progress, as a crucial stepping stone to a complete solution of a problem.
• Graduate level
• Postdoctoral level
I am also (slowly) in the process of gathering my thoughts on time management from the perspective of a research mathematician.
• Here are some general thoughts on this topic.
• Batch low-intensity tasks together to take advantage of economies of scale and to reduce distraction.
• Po Bronson’s article on the relative importance of innate intelligence versus effort.
• Fan Chung’s advice for graduate students.
• Lance Fortnow’s “Graduate Student Guide“.
• Oded Goldreich’s “On our duties as scientists“.
• Richard Hamming’s “A stroke of genius: striving for greatness in all you do“.
• Matt Might’s “Illustrated guide to a Ph.D.“
• Gian-Carlo Rota’s “Ten lessons I wish I had been taught”.
• J. Michael Steele’s “Advice for Graduate Students in Statistics.”
• Ian Stewart’s “Letters to a Young Mathematician“.
• Ravi Vakil’s “For potential students“.
• The Mathematics Stack Exchange has a number of questions and answers on career development (and one can ask further questions that have not already been posed on that site). MathOverflow similarly has questions and answers on careers. Finally, the Academia Stack Exchange has a large number of questions and answers on all academic matters, including career issues.