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Could you elaborate on what those "upgrades" look like? At least for math, I found that by around Calc II I had to start grinding problem sets to do well on ex

Could you elaborate on what those "upgrades" look like? At least for math, I found that by around Calc II I had to start grinding problem sets to do well on exams, and I can't imagine what the next "level" would look like for something like a graduate-level math course.

abdullahkhalids 4y ago
There is no concise general answer to that question, especially one that covers many fields. What you are saying is correct - from grade 1 to phd, the best way to learn is to solve problem (sciences) or create art pieces (arts) etc. As you solve problems, there are a few things I can say about how you can upgrade. * If you are level N, you should only be learning techniques at level N. If you find that you are struggling even a little bit with level N-1 techniques, adopt an immediate no-nonsense attitude about eliminating those confusions. Eg. I have seen far too many students in my second year(!) differential equation struggling with solving with quadratic equations. This means they have to constantly jump between different levels of abstraction (algebra and differential equations) and that makes the question much harder to get right. A few hours of serious review should eliminate any confusion for something that probably took 2 weeks in high school, and probably give them an entire grade bump in differential equations. * Figure out the meta-techniques at level N and take-off the training wheels. For instance, in graduate level applied math, you are working with a lot of theorems. Something that needs to become part of your study is generating positive and negative examples of each theorem as soon as you encounter it (something the book/prof did for you in undergrad). Nobody is ever going to teach you this at the grad level, as there is very little a grad prof can say to help you learn such meta-skills. But they will hit you with novel theorems in exams (or you will encounter new theorems in research), and you need to have the skill ready. * Length of each study session and intensity of your study. In high school, a smart student can watch TV and still learn everything for an exam. A undergrad can listen to engaging music while still thinking about their problem on an assignment. A grad student needs the discipline to sit in a quiet room and fully engage with the problem for several hours. A PhD student might need to think about the problem and only the problem from the second they wake up all the way to when they go to sleep. Andrew Wiles might have to lock himself away for 6 years and immerse his whole life and being into solving Fermat's last theorem. P.S. Not very happy with my answer. I need to chew on this for a few days or weeks.

abdullahkhalids 4y ago

There is no concise general answer to that question, especially one that covers many fields. What you are saying is correct - from grade 1 to phd, the best way to learn is to solve problem (sciences) or create art pieces (arts) etc. As you solve problems, there are a few things I can say about how you can upgrade. * If you are level N, you should only be learning techniques at level N. If you find that you are struggling even a little bit with level N-1 techniques, adopt an immediate no-nonsense attitude about eliminating those confusions. Eg. I have seen far too many students in my second year(!) differential equation struggling with solving with quadratic equations. This means they have to constantly jump between different levels of abstraction (algebra and differential equations) and that makes the question much harder to get right. A few hours of serious review should eliminate any confusion for something that probably took 2 weeks in high school, and probably give them an entire grade bump in differential equations. * Figure out the meta-techniques at level N and take-off the training wheels. For instance, in graduate level applied math, you are working with a lot of theorems. Something that needs to become part of your study is generating positive and negative examples of each theorem as soon as you encounter it (something the book/prof did for you in undergrad). Nobody is ever going to teach you this at the grad level, as there is very little a grad prof can say to help you learn such meta-skills. But they will hit you with novel theorems in exams (or you will encounter new theorems in research), and you need to have the skill ready. * Length of each study session and intensity of your study. In high school, a smart student can watch TV and still learn everything for an exam. A undergrad can listen to engaging music while still thinking about their problem on an assignment. A grad student needs the discipline to sit in a quiet room and fully engage with the problem for several hours. A PhD student might need to think about the problem and only the problem from the second they wake up all the way to when they go to sleep. Andrew Wiles might have to lock himself away for 6 years and immerse his whole life and being into solving Fermat's last theorem. P.S. Not very happy with my answer. I need to chew on this for a few days or weeks.