In the jobs I've liked, I do. To be more precise, I solve the problem on a whiteboard, and then type out the solution at the computer. The more experience I have, the less often I encounter gotchas while typing that I missed at the whiteboard.

The distinction matters. Are you an engineer or a tech? An architect or a laborer? Often, software developers end up being both (just as carpenters often end up designing and building). In particular, if you want your developers to be both, then you should make sure they can.