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.For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. It’s also best to keep in mind that obtaining a solution is only the short-term goal of solving a mathematical problem.The long-term goal is to increase your understanding of a subject.I just think it’s possible to achieve levels of excellence and brilliance even in your weak areas through consistent practice. After all, they made their share of mistakes and setbacks but did not give up and they were not superhuman or genetically enhanced as far as I know..That’s why I disagree with this post by astronomer Julianne Dalcanton which i found linked from your page where she doesn’t believe most people can reach the level of Feynman-Einstein-Hawking smart. And isn’t part of their fame due to circumstance and perhaps even chance-not their intellectual ability but I mean their status and the fact that their discoveries happened to be earth-shattering or were given more attention by the public at-large beyond the scientific community?And I know you say similar things on your career advice blog, and I know it’s important to be realistic and plan for graduate school and beyond I just really don’t like how people put this label of genius or prodigy on certain people to (in my opinion) make them seem able to achieve things that most other people cannot–even the levels of Einstein or Mozart.And I don’t think that’s arrogant or unrealistic.I wanted to get your honest opinion. Hey Leif, This book might be useful in pursuing the answer for your question: Disclaimer: I just read the summary and reviews of that book. I’m, at the moment, too busy with studying Maths stuff. Tao, I am a high school student, I loved math got good grades in my middle school years.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.But I find math hard and i often make many mistakes now.In fact ,i think i can work out many problems while doing my homework . Dear Professor Tao, I am a fifteen year old student currently in high school. Should I study some analysis or is it group theory that you recommend? Well, after calculus, one usually studies multivariable calculus. While trying to solve problems from my text books (like Stein’s Complex Analysis ), I notice that very often I cannot solve the hardest problems from them.