I read a great book a few years ago that gave itself this description:
For disciplines as diverse as literature, music and art, there is a tradition of examining masterpieces - the "great novels", the "great symphonies," the "great paintings" - as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.
The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem.
I found this to be a highly enjoyable way of learning math: first a problem is motivated, then it is solved. I just finished a course on algebra which was very well taught, and I learned a lot, but I often felt it wasn't well motivated - I spent lots of time learning about, say, rings, but no reasons as to why rings are useful and we should reason at that level instead of the more familiar algebra.
Anyway, this is a very long-winded introduction to my question: can anyone recommend books or (preferably) courses which follow the basic structure of: here's a problem, here's why it's important, now let's solve it?