James Polite spent much of his childhood in foster care.

In high school alone, Mr. Polite estimates, he was placed in 10 different homes. And he received little encouragement from social workers to go to college.

But Mr. Polite, now 25, still believed that college was the best next step. He found encouragement as a volunteer in his teens, registering voters and canvassing neighborhoods in New York City during Barack Obama’s first presidential bid. Mentors on the campaign trail urged him to pursue higher education.

In 2008, at a gay pride rally for Mr. Obama, Mr. Polite met Christine C. Quinn, then the City Council speaker.