Atticus may have had a few lessons in mind when he asked Jem to read with Mrs. Dubose. Firstly, it's important to note that Jem was asked to read to Mrs. Dubose when he went to apologize to her for intentionally destroying her beloved flowers (camellias). He was upset with her for insulting Atticus a bit earlier in the story. Mrs. Dubose had shouted to Jem and Scout, "Your father's no better than the niggers and trash he works for!" (ch. 11).
Jem allowed his anger to grow (instead of dealing with his frustration in a healthy manner.) He utterly destroyed Mrs. Dubose's treasured flowers, as Lee states:
"He did not begin to calm down until he had cut the tops of every camellia bush Mrs. Dubose owned, until the ground was littered with green buds and leaves" (ch. 11).
Jem later went to apologize to Mrs. Dubose, after Atticus "strongly advise[d]" him to go and talk with her. He apologized and cleaned up his mess. He explained to Atticus, "I cleaned it up for her and said I was sorry, but I ain't, and that I'd work on 'em ever Saturday and try to make 'em grow back out" (ch. 11).
She asked him to read to her, as well, when he was talking with her. This time, rather than strongly encouraging Jem, Atticus told him he would read with her:
"'Atticus, do I have to?' 'Certainly.' 'But she wants me to do it for a month.' 'Then you'll do it for a month'" (ch. 11).
I think that Atticus wanted Jem to get to know Mrs. Dubose more and to understand some of the struggles that she endured. Perhaps he wanted Jem to see more of her life so that he could understand why she might be so angry. Though Jem didn't feel sorry when he first apologized, I think that Atticus knew that Jem would grow to feel sorry when he understood more about her. Atticus wanted Jem to see what a strong and brave woman Mrs. Dubose really was. As Atticus explained to Jem at the end of the chapter,
"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. . . . She was the bravest person I ever knew" (ch. 11).