When students arrive at college these days, they hear a familiar mantra about the purpose of higher education: Find yourself. Use these four years to discover who you are. Learn flamenco dancing or ceramics, start a composting project, write for the student newspaper or delve into 19th-century English poetry. Self-discovery, they are told, is the road to adulthood.

So why is it that so many students feel such anxiety? On campus, we hear the same complaint again and again: “I’ve done lots of extracurriculars. I’ve taken a...