Is the key.
Recently, along with a couple of colleagues of mine, I have been involved in a couple of activities, in which, some of our undergrads and grad students decided to explain some of the work that is done in the Department to school children and lay public. The experience has been uniformly positive, exciting and occasionally, even, fantastic.
So, needless to say, I agree with the following quote and sentiments (expressed in this post):
Any scientist who couldn’t explain to an eight-year-old what he was doing was a charlatan.
From Cat’s Cradle by Kurt Vonnegut
As a graduate student, one of my favorite things to do is solve interesting problems. It doesn’t matter if it is a homework problem or a random question from a fellow student (honestly, I think I prefer the random questions). Lately I have realized solving problems is not the only thing that creates a good mathematician. One also needs the ability to articulate ideas and solutions in a way others can understand and relate to.
And, there is much wisdom in that Sanskrit poem which says that the complete learning consists of four parts: a quarter from the teacher, a quarter from the peers, a quarter from self-study and the final quarter by teaching. (Yes! unless you teach, your learning is not complete – a sentiment that I fanatically agree with).