*However, I don’t see why it’s not possible for me to develop mathematical abilities as strong as my linguistic abilities or even pursue a career in astronomy (which I love) or physics or even pure mathematics.*

I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies.

There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.

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.

I also have a post on problem solving strategies in real analysis. Thanks for your advice on Solving mathematical problems. [Corrected, thanks – T.] Dear Professor Tao, here are two articles on the benefits of clever note-taking for math problem solving: PS_R_A_with a strong emphasis on math competitions and Hi dear Professor Tao, I am very interested in elementary geometry and higher dimension Euclidean geometry, could you please upload chapter 4 in your problem book (I see it is about geometry), thank you very much.

I hope you are interested in elementary geometry, too, nice to meet you here! Hi Prof Tao, As an undergraduate student I often face the problem of deciding how many textbooks problems I should do before moving on, for example, Is ten questions per chapter of Rudin’s Principles of Math Analysis adequate?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.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.The long-term goal is to increase your understanding of a subject.A good rule of thumb is that if you cannot adequately explain the solution of a problem to a classmate, then you haven’t really understood the solution yourself, and you may need to think about the problem more (for instance, by covering up the solution and trying it again).Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one.Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience and hard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills.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.

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