I spend a lot of time reading physics, chemistry and biology journal articles and books both to look for interesting unsolved problems, as well as hints to the solutions to these problems. I then begin to build a mathematical model that can describe the problematic phenomenon as accurately as possible. I get advice and help from my supervisors when I am unsure of how to proceed. It is important to know what other researchers around the world think of the same problems that we are trying to solve, and research visits and conferences a few times a year help me to do that.
At work we also drink a lot of coffee, spend a lot of time on the internet and a lot of time discussing all sorts of problems that exist in the world- we are problem solvers after all!