Of all the adults Karl relates to throughout the book, Coach Gratz, his English teacher, is the one he ends up having the most complex relationship with. When we first meet him, he sure doesn't make a positive impression: He's opinionated, bombastic, and has a major attitude because he was in Vietnam. "Gratz carried Vietnam like this twenty-ton chip on his shoulder, and we all had to admire his chip" (2.99), Karl explains.
Not only does he rub patriotism and the war in students' faces a lot, he also is extremely unstable, which isn't exactly the best thing for the Madman kids, whose entire lives are based on instability. He's prone to violent, emotional outbursts in class when people don't agree with him or he really gets going on the war and then a switch flips inside him and he starts crying and apologizing. "I think that's why us Madmen loathed him," Karl says. "[It] was what a lot of us saw at home—the behavior of a father who beats his kids" (2.100).
Hang on a minute, though. It might seem like that drill sergeant from Full Metal Jacket left the service and became an English teacher, but there's more to Gratz than just the tough, "be a man" side. It's obvious in his interactions with Karl that he honestly cares about him. Karl's dad was Gratz's AA sponsor and best friend, so he seems to take a special interest in Karl because of that past friendship. The fact that he offers to meet with Karl personally so he doesn't have to go to therapy demonstrates a desire to lend him a hand.
Still, Gratz seems to keep his distance from the rest of the Madmen, even treating them with a certain amount of contempt. "You people really stick together," he snaps at them when they come to Marti's defense during the Huck Finn "two queers on a raft" debacle. "I'm not running this class based on what students think their psychological needs are" (15.130), he says, and then he sends the whole group except Karl to the principal's office.
Okay, now Gratz is starting to sound like a jerk again, but stay with us—there's a point to his abuse of the Madmen. What it seems to go back to is that he senses the tight bond between them and is reminded of how lonely he is. "Every single teacher in the school knew how tight the Madman Undeground was," he explains to Karl at the end of the book. "I looked at that letter I'd thrown away, lying in the wastebasket, and […] I found myself thinking, 'I wish I had friends as good as Karl's'" (26.71,74). Tear.
Ultimately, though, it's Gratz who becomes the responsible figure Karl needs in his life, taking him to the bank to cosign for a personal account as his "Uncle Al." He admits to Karl that he's "misjudged therapy" (26.112) and is able to see the therapy kids through Karl's eyes, casting them in a different light.