In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.
Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.
At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
> Ujiharu’s blind charges may actually have had a noble purpose. Japanese battles involving castles almost always turned into sieges, and those always ended the same way: with the nearby fields and peasant settlements being either destroyed to try and draw the lord out of the castle or looted to feed the occupying army. Some researchers believe that Ujiharu was trying to avoid a siege to save his subjects.