Why do students resort to summary so often? One obvious reason is that they've been told they have to write, say, eight pages, and they're desperate for stuff to pad out their writing. Sorry, but it almost never works: your professor won't be fooled, and will probably hold the summary against you.
But the bigger reason for summaries is a confusion over the imagined audience for an English paper. Avoiding summary has to do with the audience for your paper, and your conception of their needs.
Okay, let's be frank: English papers are a very artificial exercise. You're told to think about the audience for your papers, but you know that the real audience is one person, your teacher. Still, you should try to play along and think about the theoretical audience for whom you're writing.
You should assume that your readers are reasonably intelligent people, who've read the work you're discussing — though perhaps not very recently. That means they might need little reminders of obscure plot points, but they certainly don't need to be told the basics. “On the first page, the narrator asks the audience to call him Ishmael, and begins to recount how he first decided to become a sailor to 'driv[e] off the spleen.' He then . . .” — we don't need that.
What, then, should you recount? Anything that directly contributes to your thesis. If the evidence helps to make your case, include it; if not, leave it out. So if you want to discuss the relationship between the biblical Ishmael and the narrator of Melville's novel, you can mention the famous “Call me Ishmael” line, but if you're talking about slavery in the novel, there's no need for it.