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Biddy continues to teach Pip all she knows including an ironic little ditty about a man who goes to London and lives a fancy life. Pip continues to teach Joe everything he has learned, though he doubts Joe is taking much of the information in.
Pip wants Joe to become less common; he wants him to gain some education so he doesn't have to be embarrassed of him.
I wouldn't say Pip believes himself to be better than Joe, but he certainly doesn't think Joe is what he should be.