When Mrs. Dubose taunts the children, Jem's advice to Scout is to "hold your head up high and be a gentlemen." Why is Jem unable to follow his own advice? What does this show about him?

Mrs. Dubose is a mean old lady who lives down the road from the Finches. Wracked by pain and hopelessly addicted to morphine, she sits out there on the porch with what is rumored to be an old Confederate pistol underneath her shawl. However, that is not the scariest thing about her. She has an incredibly vicious tongue, which she uses indiscriminately against passerby. Scout and Jem come in for some particularly unpleasant abuse whenever they walk past the old woman's house. She is forever interrogating them about what they have been up to, chiding them for their supposed wildness, and telling them in no uncertain terms that they will never amount to anything.
 
Jem finds Mrs. Dubose's choice barbs particularly upsetting. However, he is a brave soul, and he is not about to show fear in the face of an ornery old woman. Atticus, as always, dispenses sage advice:

You just hold your head high and be a gentleman. Whatever she says to you, it’s your job not to let her make you mad.

On Jem's twelfth birthday, he and Scout go into town to buy some gifts. Jem buys himself a miniature steam engine and treats Scout to a spangly twirling baton. On their way home, Mrs. Dubose is out on her porch as usual, and, sure enough, she lets fly with another string of insults. Not only does she accuse the kids of playing hooky (even though it is a Saturday), she also rips Atticus for representing “trash” and “niggers.” If that is not enough, she also tells Scout that one day she will end up waiting tables at the O.K. Cafe, a notorious dive on the wrong side of town.
 
Jem's face turns purple with rage; Scout is scared. Jem, however, mindful of his father's advice, rises above it and turns to Scout:
 

"Come on, Scout,” he whispered. “Don’t pay any attention to her, just hold your head high and be a gentleman.”

 
Unfortunately, Jem proves unable to follow his own advice. Later on, in a moment of madness, he grabs hold of Scout's baton and runs up the steps into Mrs. Dubose's front yard. He proceeds to hack off the tops of every last one of her camellia bushes. For good measure, he even snaps Scout's baton in two over his knee.
 
Why did Jem do it? Why, after having shown such maturity earlier in the face of Mrs. Dubose's spittle-flecked invective, does he now finally rise to the bait? Scout herself does not have the faintest idea. The only explanation she can offer is that maybe Jem just went mad for a few minutes.
 
When Atticus confronts Jem about the incident later, he has his own explanation:

She [Mrs. Dubose] said you lawed for niggers and trash.

This has clearly been rankling Jem ever since Mrs. Dubose yelled it at him. He wanted to be a gentleman about it but could not. Jem probably snapped because he realized that, no matter how hard he tries, he can never truly live up to the gentlemanly standards of Atticus. At the end of the previous chapter, Jem said approvingly of Atticus:

Atticus is a gentleman, just like me!

This is right after Atticus shoots Tim Johnson. What Jem particularly admires about Atticus is that he does not go around bragging about his skill with a gun. He simply shot the dog because he had to. This corresponds with Jem's ideal of what a gentleman should be. Atticus came to his standards of gentlemanly conduct, not the other way around.
 
Unlike Atticus, however, Jem takes things personally; when Mrs. Dubose made those hurtful remarks about his father's law practice, something inside him started to simmer. The old lady's vicious remarks were not just insulting to Atticus, they directly undermined Jem's idealized notion of what a gentleman is. In attacking Atticus, Mrs. Dubose has also attacked that notion, a notion which is deeply personal to Jem.
 
Though naturally of a tranquil disposition, Jem has his limits—just like everybody else. Also, Jem is still just a child; he is immature in the ways of the world. It is easy for him to call himself a gentleman, but as Atticus demonstrates, you have to act like one, too. You only truly become a gentleman over time and with the benefit of acquired wisdom.