First of all, in writing a literary analysis of a short story, the writer should remain objective. To do so, one requirement is the elimination of all first-person pronouns, and replacing them with the third person. Then, in writing a literary review, or evaluation, of this story by Joyce, a consideration of its place in Dubliners may be appropriate as has been done. For literary evaluations, include the title of the work in the introduction, along with the author. (Certainly, the title of the work "The Boarding House" is significant as this environment provides Polly Mooney the opportunities of meeting a young man and snaring him.)
A literary evaluation also reflects upon some aspect of the work under question. For instance, while Joyce was at University College, he began
to fomulate his feelings toward family, church and homeland that would be played and replayed in his fiction. [enotes]
These themes are indicative of Dubliners in its mirroring of a "stages of man" in which the third "stage," or mature life after that of adoescence is played out in "The Boarding House." With the recurring motifs of the Irish paralysis, their frustrating awareness of their powerlessness to do anything about their subjugation to the British, their servility to the Roman Catholic Church, and their ability to break from an exploitive home life, this story represents very much these motifs.
With Joyce, as you have pointed out," there is a presentation of details in a method that transcends description to that of symbol," as you have also mentioned. For, as she allows her daughter to pursue Mr. Doran, whose name means exile or stranger, Mrs. Mooney becomes the parody of the religious symbol of Mary, "a little perverse madonna" as she speaks. And, she exploits her daughter in similar fashion. (Be sure to include quotations of passages that support your ideas. What you have written is fine, but it needs orgainization and suports.)