There are many features of the scientific enterprise that we can incorporate into our everyday lives. A willingness to challenge assumptions. A thirst for well-controlled data. A desire for experimental verification of ideas. An openness to observation regardless of the outcomes.
These are all well and good, and with a properly trained mind can be wielded to great effect. But there's one more feature of science that has only emerged relatively recently: the rise of the collaboration. Back in Ye Olden Times, scientists would often work solo, even going to the extreme of writing their notes using ciphers of their own devising. They would share, of course, by writing letters to colleagues or submitting articles to journals, but revelations were kept spare until results were finalized.
As the problems that scientists faced became more complex and especially as funding sources dried up, single-author papers became the exception rather than the rule. Even a purely theoretical paper using only back-of-the-envelope calculations will usually have two or three coauthors. Some papers, such as those produced by decades-long experimental efforts, will have a thousand or more names attached to it, representing the hard work of engineers, designers, technicians, coders, lead scientists, peripheral collaborators, and all their students and postdocs.
Modern science is an enterprise; almost an industry. The challenges we face require merging diverse skillsets, frequent communication, and global coordination. It's now an absolute necessity for graduate students to learn how to work within a larger collaboration. And this spirit of cooperation - that you can't solve most problems by yourself - is a wonderful thread that we can weave into all our discussions on what science is.