Firstly, I'd like to apologize for the vague title. I found it difficult to contain within a short title the scope of discussion which I'd like to address here. I hope you continue reading to understand the concerns I am laying out here.
I've been working through Spivak's Calculus, a relatively rigorous and tough book for one to work through. Naturally, I've been stuck on many of the problems and have had to look up the solutions to those problems. With each 'mistake', I try to understand the main approach/key steps presented in the solutions before attempting the problem again using the suggested approach.
However, whenever I revisit such problems a few weeks later, I often find myself completely stuck at the first step again. I get rather discouraged when this happens since it implies that I have not really learnt anything from getting through those problems.
I know that at this point many would attribute this to forgetfulness over time. But for myself, I've realised that even after spending the time with all those problems my problem-solving ability really has not improved much. (i.e I'm still only able to solve the problems I could finish with ease the first time seeing the problem) This leads me to think that there may be a better approach to working through and internalising problems.
How can I do better? Or is problem-solving ability really just a testament to the collection of 'exercises' one has stored in memory?