The Undergrad walked into my office during office hours and sat on a chair in the corner, rather than one of the chairs at my desk. I said "hello," and waited.
I looked at the Undergrad expectingly, while the Undergrad looked down at the floor in silence. We sat like that for awhile.
Eventually I asked, "what can I do for you?"
This confused the Undergrad, who didn't know how to answer.
Trying to be helpful, I said "there must be some reason you came to see me. What was it?"
"The counselors told me to go to office hours and get to know my professors."
"Why did they want you to do that?"
"So the professors will write me letters of recommendation for grad school."
I asked some gentle questions that eventually led the Undergrad to realize that sitting silently in a professor's office with nothing to say might not be a good use of either the student's or the professor's time, and might not make the best impression or lead to a helpful letter of recommendation.
Perhaps the counselors should have given clearer advice, or the Undergrad should have thought about the consequences.
A different student came to my office to discuss doing a reading course with me, having gotten the advice to take reading courses as a way to get letters of recommendation for grad school.
It turned out that the Student had only taken one or two of the standard algebra and number theory courses. I advised the Student to take more of the basic courses, before taking a specialized reading course in number theory or algebra.
The Student wanted to know about other things students could do to impress professors. I explained about Research Experiences for Undergraduates at various sites throughout the country. Upon hearing that these took place over the summer, the Student exclaimed in disgust, "Why would I want to do math over the summer?"
I was surprised and amused. "If you don't love math enough to want to do it over the summer, why do you want to get a math PhD?" I asked. "That's a big commitment."
The Student's desire to go to grad school wasn't from a love of math; it was from a lack of anything better to do.
We discussed career possibilities and courses to consider. I pointed out that mathematics opens doors in many fields of endeavor. I hope the Reluctant Student and The Silent Undergrad eventually found fulfilling paths.