addicted. The morphine has terrible side effects. Since her condition is terminal, she wants to die with dignity without the morphine. This is why Atticus says she is the bravest person he knows, and Jem learns a lesson in courage.

This is a good question. But we need to be nuanced here. What Atticus intended to teach Jem and what Jem learned might be different. 

Here is the context. Atticus had Jem go to Mrs. Dubose's house to read to her, as a punishment for destroying her flowers. So, Jem, as a dutiful son, went to Mrs. Dubose's house to read to her on a regular basis. 

After Mrs. Dubose died, Atticus pulled Jem over and told him why he wanted him to read to Mrs. Dubose. He was trying to teach Jem a valuable lesson about courage. So, Atticus explained to Jem that Mrs. Dubose tried to overcome her morphine addiction before she died. She knew it would be hard, but she did it anyway. For this reason, in Atticus's estimation, she was the bravest person he ever knew. Therefore, he wanted Jem to see what bravery really meant. 

Here are Atticus's words:

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. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew."

Jem picked up the candy box and threw it in the fire. He picked up the camellia, and when I went off to bed I saw him fingering the wide petals. Atticus was reading the paper.

Jem destroys all of Mrs. Dubose's camellia bushes, which she adores, because she is an unlikeable person who is constantly criticizing Scout and Jem and berating Atticus for defending Tom Robinson, who is African-American. To punish him, Atticus makes Jem read to her every day for a month.

At the beginning of Chapter 11, Scout narrates the following about Mrs. Dubose:

"Jem and I hated her. If she was on the porch when we passed, we would be raked by her wrathful gaze, subjected to ruthless interrogation regarding our behavior, and given a melancholy prediction on what we would amount to when we grew up, which was always nothing" (page numbers vary according to the edition).

Mrs. Dubose seems like a hateful woman, and she regards Scout and Jem as entirely unpromising. 

When Jem has to read to her, Scout and Jem regard her with distaste. Her house smells bad, and she has saliva on her mouth. She also spends a great deal of time berating the children. After Jem spends a month reading to her, his penance is over. 

Atticus tells Scout and Jem a short while later that Mrs. Dubose has died and that she was addicted to the drug morphine but kicked her addiction before she died. Atticus explains in Chapter 11: 

“She said she was going to leave this world beholden to nothing and nobody. Jem, when you’re sick as she was, it’s all right to take anything to make it easier, but it wasn’t all right for her. She said she meant to break herself of it before she died, and that’s what she did” (page numbers vary according to the edition).

Mrs. Dubose could have taken morphine until she died, but she decided to go through a great deal of pain and agony by choosing to quit. Jem's reading to her to help her pass the time and forget about her pain. Through his experience with Mrs. Dubose, Jem realizes that even people who seem detestable have secret struggles that others don't know about and that everyone deserves empathy and understanding, even those who don't at first seem likable or understandable. 

From his encounter with Mrs. Dubose, Jem learns that there is more to people than just what they show on the surface.  He also learns the value of self-control.  Mrs. Dubose is rude and cantankerous, cruel to the children and openly and vehemently critical of Atticus's defense of Tom Robinson.  Jem hates her, and destroys her flowers in retaliation of her treatment of him and his family.  When Atticus punishes Jem by making him read to Mrs. Dubose every day for a given period of time, Jem must discipline himself, bite his tongue, and be civil to the old lady.  By learning to control his impulses and show kindness even though he may not necessarily feel it, Jem takes a giant step towards adulthood.  He also discovers that Mrs. Dubose is in reality a woman of extreme courage, and he comes to respect her determined strength.  Mrs. Dubose is battling morphine addiction, and facing the challenge with tremendous bravery and tenacious will power.  

Jem had destroyed several of Mrs. Dubose's camilla flowers. For punishment, Jem was asked to read daily to Mrs. Dubose until her alarm clock sounded. Jem found himself reading more and more every day as the clock took longer and longer to actually go off.

Mrs. Dubose had been trying to free herself from her addiction to morphine. To do so, she used Jem's reading to distract her from her pain, and by the time she died, she was free. Atticus used this scenario to teach the children that courage is a value to admire and that it takes hard work to be courageous. But, in the end... it's worth it. Real courage isn't shooting things... it is doing what is right.

Jem has a blow-out of his temper when Mrs. Dubose, of whom he is terrified, enrages him by insulting Atticus (of whom she is actually a friend, with a mutual, shared respect between them). As a consequence, Jem is compelled to go to her for over a month (a week being added to the originally appointed month) to read to her while she is in a very, very ill condition. During these encounters, Jem learns courage; to control his emotions; to pursue the right course of action even when it is distasteful (or even appalling); and to turn a blind eye to what is distressing. After these encounters Jem learns, the hard way and much against his will, to respect someone with whom he deeply disagrees; to acknowledge and value courage; and to embrace as worthy a different point of view.Mrs. Dubose was a lady of fiercely staunch Old South habit, expectation and upbringing as illustrated by her tirades to Scout: “Don’t you say hey to me, you ugly girl! You say good afternoon, Mrs. Dubose!” During Jem's quite unsettling reading-aloud encounters with Mrs. Dubose, Jem has to learnphysical courage just to mount her "steep steps." As he confides in Atticus, Jem finds the Dubose home "all dark and creepy ... [with] shadows and things on the ceiling...." To forebear in the face of Mrs. Dubose' symptoms of suffering, Jem has to learnmoral courage, as Atticus tried to teach him before, by receiving her insults with bold calm and emotional tranquility:

he would gaze at Mrs. Dubose with a face devoid of resentment. ... [H]e had cultivated an expression of polite and detached interest, which he would present to her in answer to her most blood-curdling inventions.

After the encounters with Mrs. Dubose, Jem learned aboutall-encompassing courage when he understood from Atticus the determination, strength and deeply rooted courage that Mrs. Dubose needed to call up in order to break her morphine habit so that she might pass from her life and "leave this world beholden to nothing and nobody." This lesson changes Jem in a concrete way that manifests in his attitudes and behavior toward Scout, much to her confusion and dismay: "Jem hollered, 'It’s time you started bein‘ a girl and acting right!'"

The cranky, old, racist Mrs. Dubose is sitting on her porch and calls Atticus a “nigger lover” one day as Scout and Jem are passing by her house on the way to town.  When Scout and Jem return, Jem destroys Mrs. Dubose’s camellia bushes with the baton he purchased for Scout in town.  Jem’s retaliation causes Atticus to agree with her request for Jem's punishment and makes him go to Mrs. Dubose’s house each day to read to her while she is “sick” to help her through her secret struggle. In reality, Mrs. Dubose has an addiction to morphine and has vowed to break her habit before she dies.

Despite her possible symbolic representation of the dying South, Mrs. Dubose teaches Jem about taking responsibility for one’s actions (Jem’s destruction of the bushes) and that there are consequences to actions. Atticus also tells Jem that she is the most courageous person he has ever met, thus teaching them about overcoming obstacles and hardships in life.

I have always encouraged my students to analyze Mrs. Dubose as something more than just an old, mean lady.  She is a symbol of the Old South and their racist attitudes.  The addiction she is trying to break could be representative of the South’s adherence to racism. As one of the oldest residents of Maycomb, she is signaling the death of the old ways and the emergence of a new generation of people like Jem and Scout who are not racist.

There are a number of lessons learned by Jem which can be credited to his interactions with Mrs. Dubose.  Two of the most central lessons are a focus on what courage really is and on empathy.  The purpose behind Jem being forced to read to Mrs. Dubose is so that Jem can see what real courage is, not a man with a gun, but anyone who takes on a task where you know you cannot win but you give it your all anyway, as Atticus says to describe what Jem witnessed in Mrs. Dubose' last months alive.  In addition, spending time with Mrs. Dubose set up Jem to learn about empathy, the ability to put yourself in another person's shoes and attempt to see the world from their experiences, instead of judging them based on their outward behavior.

Jem actually learns two lessons. 

One: You must take the consequences for your actions. After he cuts off the tops of the flowers at Mrs. Dubose's home, Atticus sends him back to talk with her. He (Jem)  returns with the information that he is going to work in her garden every Saturday until the flowers bloom again, and he cleaned up her yard.  She also asked him to read to her every afternoon.  That was the part that upset him the most. Atticus tells him,

"Jem, she's old and ill.  You can't hold her responsible for what she says and does." (pg 105)

Later, Scout tells the readers that,

"Jem's chin would come up and he would gaze at Mrs. Dubose with a face devoid of resentment.  Through the weeks he had cultivated an expression of polite and detached interest, which he would present to her in answer to her most blood-curdling inventions." (pg 110)

However, the greatest lesson Jem learned was after Mrs. Dubose died.  Atticus, at that time, chose to tell the children that Mrs. Dubose was a morphine addict. She had taken the drug for years when it was prescribed by her doctor for pain. She could have continued its use and died without so much agony, but she wanted to be free from the drug when she died.  She had a very painful and horrible death, but died "beholden to nothing and nobody" (pg 112) .  Atticus tells Jem,

"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." (pg 112)

Atticus called her the bravest person he ever knew.  However, if you look at Atticus and his battle to save Tom Robinson, you will see that he too knew he was licked before he began, but he began it anyway, and saw it through to the end.

To Kill A Mockingbird

Jem's father, Atticus, probably answers this question best. When Jem asks his father why he calls Mrs. Dubose "a great lady" after the way she treated their family, Atticus replies, "I wanted you to see what real courage is . . . It's when you know you're licked before you begin but you begin anyway and you see it through no matter what. . . . Mrs. Dubose won. . . . She was the bravest person I ever knew." Unknown to Jem and Scout, Mrs. Dubose was a morphine addict but succeeded in breaking her addiction before she died.

There is an obvious lesson in the courage and bravery of Mrs. Dubose. Although she doesn't need to go "cold turkey", she can't bear dying being dependent on morphine, and so, at great personal cost, weans herself off this drug. Here we have another noble fight that is fought in this novel, one that is based on principles and values, and interestingly, one that Atticus Finch treats with great respect and causes him to praise Mrs. Dubose. Her struggle is perhaps a foreshadowing of the struggle for justice engaged in by other characters in the novel later on.  

Also, she reminds Jem that her Camelias cannot be destroyed unless you "pull them up by the roots". This could be seen to be a metaphorical illustration of racism - it needs to be uprooted completely for it to be eradicated.

We need to be nuanced in answering this question. 

First, Atticus wanted Jem to spend time with Mrs. Dubose to learn what true courage was. So, even though Jem did not want to read to Mrs. Dubose, he did. At the end, Atticus revealed to Jem what he could learn from her. According to Atticus, Mr. Dubose was the bravest person he ever met. The reason for this was because she wanted to overcome her morphine addiction before she died. All 98 pounds of her struggled, even though the odds were stacked against her. Here are Atticus' words:

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. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew.”

Did Jem learn this lesson? Probably in time he did. 

Second, when it comes to the rabid dog, we cannot be sure if Jem learned a lesson. On a more figurative level, the point is that sometimes people are gripped with madness, like the rabid dog. We know that we are on the right track in saying this, because the text makes this connection.

“Right. But do you think I could face my children otherwise? You know what’s going to happen as well as I do, Jack, and I hope and pray I can get Jem and Scout through it without bitterness, and most of all, without catching Maycomb’s usual disease. Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don’t pretend to understand... I just hope that Jem and Scout come to me for their answers instead of listening to the town.

Eventually, I believe that Jem learned an important lesson. Many people in Maycomb have a huge blind spot. They cannot see their racism. So, good people like Atticus are needed to advocate for what is right by challenging the madness. 

Scout learns several important lessons throughout her experience with Mrs. Dubose. After Atticus punishes Jem, Scout is rather upset that her father is making him read to their racist neighbor alone. She has a discussion with Atticus about his decision to defend Tom Robinson where she says that he must be wrong. Atticus then explains why he is defending Tom despite the community's opposition. Throughout this conversation, Scout learns the importance of following one's conscience.After Mrs. Dubose passes away, Atticus explains to his children that Jem's reading helped Mrs. Dubose beat her addiction to morphine. He then elaborates on Mrs. Dubose's courage and tells his children,

"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" (Lee 70).

Scout not only learns what 'real courage' is, but also gains perspective on the duality of human nature. From her experience with Mrs. Dubose, Scout learns that people can have both positive and negative qualities.