How do research mathematicians "do" math?

• They explore...
• They experiment..
• They engage in conversations with colleagues..
• They care about how and why..
• They use creativity and intuition..
• Initially fuzzy defs/theorems get refined..
• Pictures, diagrams..
• Groupwork (collaboration)..

...not exactly "read-text/listen-to-lecture/do-problem-set"

What is reasonable for our students?