What is the role of education in Sense and Sensibility?

Lucy was naturally clever; her remarks were often just and amusing; and as a companion for half an hour Elinor frequently found her agreeable; but her powers had received no aid from education: she was ignorant and illiterate; and her deficiency of all mental improvement, her want of information in the most common particulars, could not be concealed from Miss Dashwood, in spite of her constant endeavor to appear to advantage. Elinor saw, and pitied her for, the neglect of abilities which education might have rendered so respectable; but she saw, with less tenderness of feeling, [Lucy's] thorough want of delicacy, of rectitude, and integrity of mind ...
Jane Austen's Education
Biographies point out that Jane was sent away from home to be educated, although it was not for an extended number of years. Jane's own education was begun at home. Her father was a scholar and her mother was educated. With her father's extensive library, Jane could expand her formal education through her own readings. It is true that Jane, along with most women of her day, did not learn the classical languages of Greek and Latin, so even as the daughter of an eighteenth century Oxford proctor, her education was limited.
Education in the 1700s
The first thing that it is important to grasp is that prior to the eighteenth century (1700s), the number and variety of English dialects made a uniform language and, as a consequence, a uniform education impossible. It was only during the seventeenth century (1600s) that a "received English" with uniformity of grammar and vocabulary began to unite the multitude of dialects across the country; at least this was true in the upper classes where education was already somewhat prevalent. Although standardized English was shifting a standard grammar and vocabulary had come into existence. In addition, a standard for uniform education had come into existence. This is seen in the changes to public opinion evident toward education for both boys and girls that came to the fore in the eighteenth century (1700s).
Development of Education 1500–1800
With the closure of Catholic convent and church schools in the 1500s, grammar schools, funded by privately contributed endowments, were opened. The 1700s saw the beginning of charity schools where girls and boys were taught the needed skills to conduct themselves and to engage in productive work.
The changes in England in the 1500s in educational philosophy and practice because of the support given by humanists to higher education, meant education was evolving rapidly. Protestant reformers and Elizabeth I advocated for education for all children. Despite the slowing of educational enthusiasm as Puritanism grew more popular, the changes to educational philosophy and practice continued to gain ground through the 1600s and 1700s, with the greatest advances coming in the 1800s but occurring after Jane's death.
Upper Class Education
Throughout the 1700s, the children of the three levels of the upper classes began their educations by being tutored at home by private tutors for the boys and private governesses for the girls: The girls most often continued their entire educations with their governesses, as Emma did in Austen's Emma, while the boys went into professional apprenticeships or university. In Pride and Prejudice, Lady Catherine de Bourgh reinforces this idea when she, in shock, chastises Elizabeth because Mrs. Bennet had kept no governess for her five daughters.
The major differences between how boys and girls were educated were two: boys, but not girls, studied the Classical languages of Greek and Latin, which opened all avenues of study since most academic and scholarly texts had been and still were written in or translated into Greek and Latin; boys learned skills that could translate into earnings and spanned fields from the law to farming, depending upon the social class of the boy studying; girls were educated in subjects that did not require classical languages, like geography, history and maths, while learning the arts or domestic skills, depending on the girl's social class.
For girls in the upper classes, their education was significant enough that their logical thought, creative insights and communication powers were excellent, especially if you take writers like Jane Austen, Elizabeth Barret Browning and the Bronte sisters as educational trend indicators.
Girls' Education in Sense and Sensibility
Austen herself offers a very good implied example of what a girl's education might have been in the 1700s through her own fiction, voice and logic. She also presents very good comparative examples in Sense and Sensibility of what a girl's education might have been in this era through the juxtaposition of Elinor and Marianne; Elinor and Lucy; Mrs. Dashwood and Fanny Dashwood; Mrs. Dashwood and Lady Middleton; Mrs. Jennings and Mrs. Ferrars. Lucy is of the lower class, the fourth rank of class structure, and her language and thought patterns reflect this.
[H]er powers had received no aid from education: she was ignorant and illiterate; and her deficiency of all mental improvement, her want of information in the most common particulars, could not be concealed...
A sound education for girls was a real possibility for girls in the upper classes, although such a possibility decreased as the class rank decreased. Education in the upper classes could yield women with admirable minds like Elinor and Jane Austen herself.
Boys' Education in Sense and Sensibility
Edward's education is directly contrasted to Robert's when Robert and Edward, both, discuss the demerits of private tutelage, such as Edward received, and the merits of public education, such as Robert received.
The results of the educations of Sir John Middleton and John Dashwood are contrasted against the results of education shown in Colonel Brandon, while those of Mr. Palmer and Willoughby are contrasted against those of Colonel Brandon also. Austen thus illustrates the thematic point that even among men, the educations are varied and produce varied results, notwithstanding the learning of Greek and Latin.