Why did Jem destroy Mrs. Dubose's flower garden? Was Jem's punishment fair and appropriate?

Jem destroys Mrs. Dubose’s flowers because he is angry that she called his father names and he was punished by having to read to her.

When Jem attacked Mrs. Dubose’s flowers, he was reacting to her racism and the town’s.  Most people did not approve of his father defending a black man.  Jem had been able to keep his head for the most part, but he just lost it with Mrs. Dubose.

Scout gets upset when Mrs. Dubose begins to yell at them.  She insults them for being on the street when they should be in school, even though it’s Saturday.  She also scolds Scout for wearing overalls. Finally, she turns on their father.  Most of the town disapproves of his defending Tom Robinson because he is a black man accused of raping a white woman.  Some of them take it out on his children.

“Yes indeed, what has this world come to when a Finch goes against his raising?

I’ll tell you!” She put her hand to her mouth. When she drew it away, it trailed a long silver thread of saliva. “Your father’s no better than the niggers and trash he works for!” (Ch. 11)

Jem told Scout to keep her head, but this was too much for him.  He took her baton and attacked Mrs. Dubose’s flowers.  They are the one pretty thing about her, and their presence seems to contradict her words.  Basically, he just wants something to lash out at, and the flowers are there and defenseless.

He did not begin to calm down until he had cut the tops off every camellia bush Mrs. Dubose owned, until the ground was littered with green buds and leaves. He bent my baton against his knee, snapped it in two and threw it down. (Ch. 11)

Scout panics because her brother usually keeps his head.  She worries about what Atticus will say or do.  His solution is to send Jem and Scout over to read to Mrs. Dubose every day for a period of time.  She sets an alarm and he reads to her until it goes off.

It is not until later that Jem learns why he was sent there.

I wanted you to see something about her—I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do. (Ch. 11)

Atticus wanted them to understand that there is more to people than there may first seem to be.  The punishment may seem harsh, but it was intended to help Jem get a more nuanced understanding of Mrs. Dubose and why she was acting the way she was.  She was addicted to morphine, and that made her more vicious. 

Atticus wanted Jem and Scout to understand that sometimes people have to fight for what they believe in, because that is what he is doing with Tom Robinson.  It is not always easy, but it is necessary and worth it in the end.  It is a lesson in the complexity of human nature that young Jem needs to learn.