Responding to Gaffa (I kind of intended to respond right after the comment, bot got sidetracked):
When approaching a scientific or mathematical problem, I often find myself trying hard to avoid having to calculate and reason, and instead try to reach for an “intuitive” understanding in the back of my mind, but that understanding, if I can even find it, is rarely sufficient when dealing with actual problems.
I would advise you to embrace calculation and reason, but just make sure you think about what you are doing and why. Use the tools, but try to get an intuitive understanding of why and how they work, both in general and each time you apply them. It is true that formualaic rules can serve as a crutch to avoid the need for understanding, but if you throw away calculation and reason, you are not likely to make much progress.
Finally, be realistic in your expectations: for a complicated problem, you should not expect to be able to get an intuitive understanding of the solution as a single step, but you can aim for a chain of individually intuitive steps and, if the chain is sufficiently short, an overall intuitive understanding of how the steps relate to one another.
Responding to Gaffa (I kind of intended to respond right after the comment, bot got sidetracked):
I would advise you to embrace calculation and reason, but just make sure you think about what you are doing and why. Use the tools, but try to get an intuitive understanding of why and how they work, both in general and each time you apply them. It is true that formualaic rules can serve as a crutch to avoid the need for understanding, but if you throw away calculation and reason, you are not likely to make much progress.
Finally, be realistic in your expectations: for a complicated problem, you should not expect to be able to get an intuitive understanding of the solution as a single step, but you can aim for a chain of individually intuitive steps and, if the chain is sufficiently short, an overall intuitive understanding of how the steps relate to one another.