phd - To what extent should a professor give students freedom to do independent research?

During the discussion How/when to become independent in research as a graduate student?, there were one or two professors who not only agreed with the idea of independent research, but also said that they encourage their PhD students to conduct independent research as long as this helps the students to be more mature and successful researchers in their future academic career. I decided to bring this question to the main topic that if you are a professor and allow your students to do independent research:

• What do you mean by independent research?


• Should your students conduct independent research and publication in the field of their PhD/MSc research topic or it can be in any other field of their major and interest?


• If their independent research is not in the field near to their PhD research, should they inform you about their other research projects or not?

Answer

I'll answer, not from the perspective of a professor (I'm not one) but as a student who had advisors who were very generous in that regard:

• What do you mean by independent research?

When people say this, they usually mean research that is outside a clear plan of action. For example, in the biomedical sciences, it's somewhat common to turn a particular specific aim of an R01 grant into a PhD student's dissertation. That's a plan. But along the way, the student may be interested in a methodological musing, a side project, a short data analysis task on something interesting that's come up, etc. So basically, I'd take it to mean anything that's outside what you've been tasked to do, or come up with tasking yourself to do, as part of your degree program.

• Should your students conduct independent research and publication in the field of their PhD/MSc research topic or it can be in any other field of their major and interest?

It probably shouldn't detract from their research topic, but I've never seen any reason why more general topics are a problem. Indeed, if the student is restricting it to their research topic, than why isn't this part of their thesis/dissertation in the first place?

• If their independent research is not in the field near to their PhD research, should they inform you about their other research projects or not?

It's good for them to inform you of what they're working on regardless. They may be able to provide support, put in a kind word to an editor, etc. And these people will be writing your letters of recommendation - they should be able to talk about the exciting stuff you do even if it's not directly their project. And they should be informed so they can tell if you're potentially overburdening yourself.