This is a rather broad (and perhaps too philosophical) question about undergraduate and graduate mathematics education.
Gowers, in his article "the two cultures of mathematics", observes differences between the problem solving and theory building aesthetics in mathematics. To slightly perturb an analogy of Alexander Grothendieck, one might view the problem solving culture as focusing more on a "Yang" or "attacking" mathematical aesthetic, and the theory building culture as focusing on a more "Yin" or "yielding" aesthetic. Of course, I have taken liberties in my perturbation of this analogy, and the distinction is probably best understood by reading Gowers's article linked to above.
Of course, the whole of mathematics involves an interplay of the "Yin" and "Yang" of the two cultures, and every mathematician is at any moment looking at things more as a problem solver than as a theory builder or the other way around. Subjects emphasizing the different styles mostly take care of sorting mathematicians into appropriate communities for their tastes, but I can't help but wonder if there are better ways to recognize the diversity of mathematical tastes in my classroom.
Some students simply love to read and learn about mathematics, and gain a very strong base of ideas to think about, and there is a danger of these students becoming depressed by seeing a problem solver work incisively on a problem.
Some students are excellent at problem solving, but become depressed when looking at classmates who have a broader knowledge base. Many such clever students are turned off by routine mathematics classwork aimed at building up their knowledge base.
The above perceptions extend rather far into one's career. Once, a problem-solver friend of mine was shocked when he overheard a discussion between a few mathematicians (one of whom had a Fields Medal) and noticed that the participants didn't have a solid grasp of a certain definition they were talking about. (I think it might be common among some theory-building types to try to tease new perceptions from an ill-defined understanding of a definition.)
On the other hand, a former graduate student of an IMO Gold Medalist who is now primarily a theory-builder told me stories about the disdain his advisor had for results that were "just combinatorics".
What concretely can we do to better embrace the diversity of mathematical tastes in our departments? What can we concretely do as mathematics educators to encourage communication between the two cultures?
(EDIT: Here is an example of the sort of thing I'm hoping to see, ideas for concrete pedagogical practices that encourage either "culture" or perhaps how to concretely achieve a balance.)