How is Jane Austen's novel Emma an education for its heroine, Emma Woodhouse?

Emma, like many young people, thinks that she knows everything, especially what is best for the important people in her life.  Sometimes Emma's motivations are altruistic, that is, she wants what is best for another with no regard for herself.  Unfortunately, Emma frequently moves people around like chess-pieces, pawns in the game of matchmaking, staged for her amusement.  Emma claims early success in creating a "good match" for her governess, Miss Taylor to Mr. Weston.  Like a gambler, Emma gets a certain high from her victory with the pair and wants the adrenaline to continue.

Her plans are messed up when she attempts to match her friend Harriet, who (in Emma's opinion!) has had the audacity to select her own mate; and, to make matters worse, a man Emma does not approve of. 

Emma's plan to turn Harriet instead to the charming Mr. Knightly backfires when the plan works, for now Emma herself has fallen in love with Knightly.  And so has Harriet. 

Emma grows because she learns that people are not pawns, that she is not in control, and that she can give up the "addiction" and learn to live with less manipulation of others (even if she is not totally cured.)  This is the "education" of Emma. 

In Jane Austen's Emma, how does education affect the novel?

Although the novel's opening sentence describes Emma as "handsome, clever, and rich," the novel shows the problems that arise from her lack of a good education. Emma's mother died when Emma was very young. Miss Taylor, who marries and becomes Mrs. Weston as the novel opens, was Emma's governess. A kind companion, she never was able to enforce discipline, so Emma is half-formed. Emma, for instance, envies Jane Fairfax's skill at the piano, but doesn't have the self-discipline to apply herself to the art. As Mr. Knightley says of Emma, she is always drawing up reading lists, but not actually doing the reading. He says to Mrs. Weston:

I am done with expecting any course of steady reading from Emma. She will never submit to anything requiring industry and patience, and a subjection of the fancy to the understanding.

The novel shows that the lack of self-discipline that has kept Emma from reading and learning in a systemized way leads her to jump to conclusions based on "fancy" and not on solid facts. Emma's half-baked education has bad effects on the people around her, for she misunderstands people's motives and takes on as her companion Harriet Smith, a pretty young woman who reads romance novels and has "scrambled" her way into a "a very indifferent" education at Mrs. Goddard's school, but who is not likely to elevate Emma's mind. Emma's "fancy" leads her to jump to false conclusions, such as that Mr. Elton is in love with Harriet, creating the madcap confusion about love that gives the novel its Midsummer Night's Dream quality.

As Mr. Knightley points out, Emma "is spoiled by being the cleverest of her family." At ten, he says, she could "answer questions which puzzled her sister at seventeen. She was always quick and assured . . ." But without a solid education, this intelligence is frittered away and misused.

Jane Fairfax, on the other hand, was taken from Highbury at a young age and raised by her father's friend Mr. Campbell. She has received "an excellent education." Emma admires and envies her for her "elegance." Jane is discreet and doesn't matchmake and meddle as Emma does.

The novel as a whole condemns the improper education of rich young women like Emma, who spend their time drawing, reading novels, keeping books of "charades" and flitting from one activity to another while becoming good at none. Emma is bored and this leads to trouble. It's better, Austen implies, to provide a solid and structured education for women, so that they have resources to fall back on other than meddling in other people's lives.