How to improve mathematical thinking? I am a sophomore in high school. When I approach a math problem or concept, I normally try to determine a formula or systematic method of solving all similar problems. However, since I entered precalculus, I have found that this approach does not work. I really love math and find it interesting and exciting, but I find that I am unable to solve the kinds of conceptual problems that I have been doing more recently. For example, when I try to do a counting problem, it is often unclear how to approach it. How can I improve my problem-solving and critical thinking skills? Also, how can I approach problems logically to solve ones that are not clear at first?
 A: I am a PhD student of pure mathematics and I just can talk from my experience. I think that mathematical thinking can be improved with your experience solving problems and reading. There are 2 options. 
Option 1: improve your mathematical thinking by yourself. This means try to approach to the problem from all the possible points of view that you can imagine. Organize them, try to apply them one by one and draw a lot. Try to improvise and start solving similar problems in simpler versions. This is very hard to do, specially when you learn in a systematic way, because this require creativity but if you spend time doing this even if you don't solve it you can grow a lot and develop intuition.
Warning! Do not spend more than 1 week in the same problem. Not all the people solve problems quickly and that is just fine. If you cannot solve a problem after your hard work is good idea  to ask for help (books, mentors, internet, etc) or just leave it in a special list and go forward. You will be able to solve that list in the future. Reading some comments reminds me that teaching to others the exercises that you can solve is a very powerful way to improve your mathematical thinking, it helps to organize your knowledge and disciple your mind. Please be patient with others. Some day someone will be patient with you and you will need it!!!
Option 2: improve your mathematical thinking using help. Some people may think that  asking help to others or books destroy your creativity and limit your mathematical thinking to the creativity of others. But only few gifted can afford that. I think that there is nothing wrong gathering some strategies of others to enrich your own bunch of tools. Consult a friend, professor, books or forums like this. When you ask for help the method to resolve a problem is something that you haven't thought, however the new experience can help you to solve new problems in the future. When you are facing a new kind of problem and you do not have a clue where to start,  look for examples and solved exercises. If you are in high school most likely there is a lot of reading material, examples and solved exercises for the topics  that you are interested.
At the end your experience resolving problems and your background will develop your mathematical thinking, and you can do it using the options 1 and 2.
