When the fall term begins at Nazareth High School today, one teacher will be conspicuous by his absence.

Matthew Greene, a popular math teacher at the school 65 miles north of Philadelphia, vanished while on a backpacking trip in California last month. Police say Greene was there to hike and climb the Eastern Sierra, a region that features extremely steep and rugged terrain.

His disappearance has baffled friends and family who call Greene a highly experienced, cautious outdoorsman.

Nearly a month after he was reported missing, Greene’s family was offered a tantalizing clue as to his possible whereabouts: A hiker on Friday found a pair of eyeglasses that look “strikingly similar” to the ones worn by Greene, his sister Tiffany Minto said. The glasses — found hanging on a sign at a trailhead near the Inyo Craters region — have been turned over to the Mammoth Lakes Police Department, and Greene’s optometrist has provided information about his prescription in hopes of helping police make a match.

Greene, 39, had arrived in late June to hike and climb with friends. But when his Subaru blew a head gasket, they went on with their trip while he stayed behind at a campground in Mammoth Lakes to wait for repairs. The last time anybody heard from him was July 16, when he chatted with his parents back in Pennsylvania, called the repair shop and traded text messages with a friend.

Some of his gear was discovered to be missing, and pages were torn out of his guidebook, leading to speculation that he might have headed in that direction for a hike.

Several of Greene’s friends flew to California and canvassed bus drivers, store owners, librarians and others in Mammoth Lakes, posted fliers at trailheads, checked summit registers and spent several days in the back country looking for clues. The search has gone airborne, too. But they turned up no trace of the former Peace Corps volunteer. And, until Friday’s recovery of the glasses, police had been struggling to develop leads.