Jasmine Miller, who grew up in Tennessee and graduated from Harvard in 2013, has some illustrative anecdotes to explain how low-income students at elite colleges get subtle and not-so-subtle reminders that they aren’t like their classmates. An Ivy League economics professor, for example, might try to elicit a discussion by asking, "How many of you were raised by nannies?" And a low-income student is bound to get a party icebreaker like "Where’s your favorite place to go abroad?" Ms. Miller provided those anecdotes to give perspective to a select group of elite-college presidents and high-level administrators during a symposium here on Thursday at the headquarters of the Jack Kent Cooke Foundation, which gives what it calls the nation’s largest academic scholarships to roughly 250 high-performing, low-income students each year.