PHILADELPHIA—On a quiet Sunday afternoon at the University of Pennsylvania, a dozen students sat in a circle, turned to one another, and asked: “Are you thinking of killing yourself?”

The difficult practice, accompanied by uncomfortable giggles and prolonged eye contact, came toward the end of a four-hour training session in a technique known as active listening. Students are taught to ask the question, among others, with a gentle and direct tone. The method and question can help reduce the risk of suicide, training experts say.

With 14 student suicides in the past six years, this Ivy League university has been asking hard questions and has bolstered its mental-health resources. But the recent death by suicide of a high-profile mental-health administrator—Gregory Eells, executive director of Penn’s Counseling and Psychological Services program that provides therapy sessions for students—highlighted the complexity of the school’s continuing battle against suicide.

“On a symbolic level, Dr. Eells’s death hit harder. Because of his position, it’s a stronger message across the university than I think student deaths are in a weird, kind of bizarre way,” said Greg Callaghan, president of the Graduate and Professional Student Assembly who had met with Dr. Eells about graduate student mental-health initiatives.

The U.S. suicide rate for nearly all ages increased from 1999 to 2017, according to the most recent figures from the Centers for Disease Control and Prevention. It is the second-leading cause of death among Americans aged 10 to 34. Suicide has touched college campuses around the country, and school administrators have responded with increased support services. Students have also gotten involved in suicide prevention.