the fact that you graduated high school and university and earned your doctorate so young), I’m wondering if you think you possess something that only a few others have in terms of intellectual ability or not.

I have always believed that if someone applies himself and puts in enough time, effort, concentration, and perseverance they can accomplish whatever they set their mind to.

to expand out the definitions, solve some special cases, and isolate key difficulties) is also a very important measure of progress (see this previous post of mine on this topic), as is the practice of constantly asking yourself “dumb” questions in the subject (as discussed in this post).

One should also not focus on the most difficult questions, but rather on those just outside your current range.

Solving homework problems is an essential component of learning a mathematical subject – it shows that you can “walk the walk” and not just “talk the talk”, and in particular identifies any specific weaknesses you have with the material.

It’s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.

For instance, if after failing to solve a problem, you receive the solution and study it carefully, you may discover an insight or problem-solving technique that eluded you before, and will now be able to solve similar problems that were previously out of reach.

One should also bear in mind that being able to partially solve a problem (e.g.

As you are still several years away from having to attack research-level mathematics problems, your current skill in solving such problems is not particularly relevant (much as the calculus-solving skill of, say, a seventh-grader, has much bearing on how good that seventh-grader will be at calculus when he or she encounters it at the college level).

The more important consideration is the extent to which your problem-solving skills are improving over time.

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