Which single verb describes the mission of universities?

Think!

Research is all about thinking about the world we live in; whether it is genetics, cosmology, literature, engineering, or economics, ...
Reality is stratified and one observes different phenomena in different systems. As a result, one needs to think in distinct ways in order to develop concepts, laws, and methodologies for each stratum. 
Note that thinking is central to experiments: thinking how to design the experiment and apparatus, and how to analyse the data produced and relate it to theory.
This is why we have disciplines. Each discipline involves a disciplined way of thinking.

Teaching is all about helping students learn how to think.
For specific disciplines, it involves learning how to think in a particular way.
Thinking like a condensed matter physicist is an art to learn.
Similarly, thinking like an economist is a unique way of thinking.

If this is the mission of modern universities are they successful?
On one level they have been incredibly successful.
Almost all the disciplines and knowledge we have were created in universities.
These ways of thinking have been incredibly productive and revealed things we might never have anticipated or dreamed of. Whether it is the genetic code, quantum field theory, game theory, or studies of ancient histories and cultures, ....

Furthermore, universities have really taught many students to think critically and creatively, not just about academic matters. University graduates have used their thinking skills in constructive ways, whether in inventions, starting companies, journalism, politics, philanthropy, ...
It should be acknowledged that this education does not just occur in the classroom but in informal contexts and involvement in student clubs and societies.

However, when you consider the resources that have been expended globally, both in teaching and research, you have to wonder whether universities are now failing at their mission.
This is reflected in a sparsity of critical thinking on many levels and in many contexts.

In the Majority World, universities try to mimic Western ones, at the superficial level of structures and curriculum. However, largely they focus on rote learning and not questioning teachers. This tragedy is captured with humour in my favourite Bollywood movie scene. Not only are students not taught how to think, they are actually taught not to think at all!

Yet, Elite universities now have a lot to answer for. The administration has become decoupled from the faculty and so we have metric madness and mindless marketing. Many of the statements or decision making processes (e.g. ignoring uncertainties, listing journal impact factors to 4 significant figures or cherry picking data to enhance the "ranking" of an institution) would be not be allowed in a freshman tutorial or lab.
Yet faculty are not without fault. Critical analysis will be avoided if publishing in a luxury journal is on the horizon. Then there is the hype of faculty about their latest research, whether in grant applications or public relations.

There are countless other ideas about what the mission of the university should be: training graduates for high paying jobs, wealth creation, enhancing national security, elite sports, industrial research, creating good citizens, ...

Many of these alternative missions are debatable, but regardless, they should be subordinate to the thinking mission.

Key to the thinking mission is academic freedom. Faculty and students need to be free to think what they want about what they want (within certain civil and resource constraints). Political interference and commercial interests inhibit such thinking.
It is interesting that Terry Eagleton, considers that the primary mission of universities is to critique society.

I thank Vinoth Ramachandra for teaching me this basic but crucial idea.