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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).For related reasons, one should value partial progress on a problem as being a stepping stone to a complete solution (and also as an important way to deepen one’s understanding of the subject). Note: My English is quite poor, you may experience this in the solution.The more important consideration is the extent to which your problem-solving skills are improving over time.
Does that mean you don’t consider yourself a genius or you don’t really see a distinction between yourself and others who apply themselves and are ambitious?
I am not saying I don’t believe some people are naturally more gifted at certain things or develop stronger skills or have stronger talents than others.
But, if you put the problem into a calculator or Google search, the slash would be interpreted as a division symbol, which would change the order of operations used to solve the problem, producing a different answer. Watch the video below, which explains the common mistake that is made and how to correctly solve the problem.
Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one.
Surely some of the scientists working today will make equally groundbreaking or insightful discoveries or develop innovative theories and thus can fairly be labeled “genius” or as having the same level of smarts? I think I have developed a stronger aptitude for language than for math and due to suffering from depression in high school and middle school I didn’t push myself nearly as much as I could and lost much of my motivation.
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 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?
The more problems I do on a specific topic the slower it takes to reach graduate level mathematics. Tao: I hava translated this essay into chinese, I’m sorry I couldn’t translated it well enough, as my ability in english is as poor as mathematics.
But i am very nervous during my math exams and i almost forget everything i have learnt. I am currently self-studying some non-rigorous calculus. Try to see if you’ve learned everything in regular calculus, and then go onto . Since research is about hard problems, does that mean I don’t have what it takes to be a mathematician?
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).