What do Mathematicians do?

After I heard someone ask about what a mathematician does, I myself wonder what it means to do mathematics if all what one can answer is that mathematicians do mathematics. Solving problems have been considered by some as the main activity of a mathematician, which might then be the answer to the question. But, could reading and writing about mathematics or crafting a new theory be considered as serious mathematical activities or mere extracurricular activities? It seems mathematicians and mathematics students are expected to always solve problems. Many mathematicians I know or hear about have always been busy with solving problems; some of the named theorems, such as Heine-Borel Theorem or Brouwer Fixed-Point Theorem, could have been problems on which Heine, Borel, and Brouwer spent a long time to solve. Also, many papers published in journals are solutions to problems, which makes me think that mathematical research is synonymous to solving problems. I also have the impression that solving problems is seen by some as the “right” way to do mathematics by the emphasis put on such activity. But, if someone writes an expository paper about a mathematical topic, could that also be seen as doing mathematics? Could this enterprise be counted among a mathematician’s achievements? For students being trained in mathematics, should there also be an emphasis on reading and writing about mathematics, which is not necessarily reading and writing proofs? Furthermore, when a mathematician comes up with a theory after years of work, could he be seen as having wasted his time while he could have solved many problems in the time he took him to complete his work?  If one thinks it indeed is a waste of time, would one also see problems arising from this theory as un-mathematical since their source is not mathematical enough? What if this theory has been inspired by a problem, would that qualify it as a genuine mathematical activity? Because of this apparent ambiguity about what constitutes the “right” mathematical activity, what would students of mathematics need to focus on for their training? Do they need to focus more on solving problems, which necessarily could involve the other activities since understanding a problem might demand some preliminary reading about its origin and motivation and solving it might push one to write about it for a wider audience and inspire one to come up with a new theory in the process of solving the problem? So, what would it mean to do mathematics? Besides the activities aforementioned, would there be other activities considered as doing mathematics? Please, share your comments.