Developing abilities to create, inquire into, qualify, and choose among mathematical problems is an important educational goal. In this paper, we elucidate how mathematicians work with mathematical problems in order to understand this mathematical process. More specifically, we investigate how mathematicians select and pose problems and discuss to what extent our results can be used to inform, criticize, and develop educational practice at various levels. Selecting and posing problems is far from simple. In fact, it is considered hard, complex, and of crucial importance. A number of criteria concerning personal interest, continuity with previous work, the danger of getting stuck, and how fellow mathematicians will respond to the findings are considered when mathematicians think about whether to approach a specific problem. These results add to previous investigations of mathematicians' practice and suggest that mathematics education research could further investigate how students "select and develop" problems, work with "multiple problems over a longer period of time," and use the solutions to problems "to support the development of new problems." Furthermore, the "negative emotional aspects of being stuck" in problem solving and students' "conceptions of solvability" and "relevance of or interest in" a mathematical problem are areas of research suggested by our study.