An in person dialogue between me and a (among other things) mathematician about doing research your own way. The conversation is not recalled word for word, but aims to reflect the main ideas we discussed.
D: What are your favorite types of math books?
2: I like books that contain a lot of problems without much additional theory, because after solving them you create your own picture of what the important aspects of the topic are. When reading a text book, I focus on learning the key ideas rather than working through all the results one by one.
D: Does it not take a lot of time working through such a book?
2: In the beginning, yes, but you get better with practice.
D: How do you think our knowledge in mathematics should evolve; is it better to delve deep into certain areas (pictured as stacking blocks on height) or try to get a wider picture by exploring new directions from scratch (like placing blocks next to each other on the ground) and focusing less on what's known?
2: Personally, I like to build the groundwork rather than working on height, but it is good have people doing both kinds. Personal taste plays a role; however I think there are too many people crammed at the top at the moment. We need to connect different fields by building a steady base between them on the ground, not just by building foot-bridges at the top.
D: Isn't it much more difficult to stay on the ground? In order to discover something truly "fundamental", don't you also require a bit of luck?
2: I don't think there is a matter of "luck"; I managed to find "fundamental" results of different flavors three times in a shorter timespan!
D: Why do you think there are so few people coming up with such results then? Do you need to be exceptionally talented?
2: I think it is because people don't know that they can, and therefore never try working this way.
D: But, if you do research without spending much time on reading the theory that we have established, won't you be more likely to reinvent the results we already have, rather than coming up with something new?
2: You will certainly reproduce many results, but this way you will both learn much more than by just reading it. You will not only understand the key ideas in your field much better, but you might also taking these ideas in a different direction. Slight deviations from the usual ways of looking at things can often open up new directions.
D: How do you become good at problem solving?
2: You focus on different techniques, one at a time - this gives the most rapid improvement! Try solving problems together with friends, for instance.
Idea: Instead of New Year's Resolutions, wouldn't it be better with new week's resolutions? Try to focus on something new each week, say - practicing/learning something - wouldn't it be more fun, rewarding, and perhaps easier to keep up since you can adapt the resolution to what suits you best that particular week?