Joel disappeared with his brother, William Hoag, and a friend, Edwin Craig Dowell. The boys were last seen on the south side of Hannibal, Missouri between 4:20 and 5:00 p.m. They were carrying shovels and a flashlight, and said they were going to explore Murphy's Cave, near the Highway 79 construction project. Joel, who was interested in snakes and geology, had been punished the night before for going to the cave without permission. The boys have never been heard from again.

That day, the construction work resulted in a cave-in at Murphy’s Cave. Because of the circumstances, authorities theorized the three boys were trapped inside the cave when it collapsed and weren't able to get out. Extensive search-and-rescue efforts turned up no sign of any of them, however. In 2006, construction workers building a new elementary school on Highway 79 found a small cave about 50 feet long. Authorities checked it to see if the Hoag brothers or Dowell were there, but they didn't find anything. Because nothing was ever found, there's speculation that the children weren't lost in the cave and something else caused their disappearances. There's no evidence to support any other theories, however.

William and Joel left behind five brothers and four sisters, and Dowell left three brothers. Joel and Dowell were both eighth-graders at Hannibal Junior High at the time of their disappearances. The three boys' cases remain unsolved. In 2010, author Charles Stewart published a book about them, titled A Sorrow of the Heart.