In what ways, specifically, is Miss Moore an outsider in her own community? Which of her beliefs, attitudes, and behaviors distance her from the children? From other adults in the community? Is she able to overcome these barriers to full communication with her students and with Sylvia in particular?

There are many reasons why Miss Moore stands out in the community. Right away, we see she has "nappy hair, and proper speech and no makeup," suggesting that other people in the neighborhood have quite the opposite. She is much more "proper" than others in the neighborhood, and the text tells us that she stands out as well because she is dark-skinned. She also has a college education and is concerned with the children's education, something not a lot of people in the community care about.
She is distanced from the kids because she isn't as poor as them and doesn't seem to understand what their life is like. She tells them that they live in slums. They can't relate to her appearance or success. The parents also feel distanced by this, because the feel they need to impress her by making the kids look better or by doing extra work on their appearance before they see her. She makes them feel that their everyday look isn't enough.
It's difficult to say whether or not Miss Moore was able to overcome the barriers and reach Sylvia. She tries to show them the "other side" and help them see that they are poor and help motivate them to want to change the systems of economic injustice. It's true that Sylvia is more determined by the end of the story, but she's still very bitter toward Miss Moore and doesn't seem to have accepted the older woman's ideas quite yet.

Miss Moore is considered an outsider in "The Lesson" due to her college degree and "proper" sense of decorum, which distinguish her from other inhabitants of the neighborhood. The first paragraph reveals that her "nappy hair and proper speech and no makeup" bring about the children's laughter and hatred. Her speech patterns and hospitality gestures, including gifts of "some sachet she’d sewed up or some gingerbread she’d made or some book," are seemingly unappreciated when shared on neighborly visits. Although the adults gossip about her behind closed doors, they nonetheless allow Miss Moore to enrich their children's experiences with outings and instruction, as she considers herself responsible for their education. The children are regularly required to dress up for and devote free time to Miss Moore's lessons, which they find particularly irritating.
On the day in which the story takes place, Miss Moore takes a group of children from their poor neighborhood to ritzy Fifth Avenue in Manhattan. Her talk on wealth distribution and inequality is initially met with ridicule, but the children better understand the concept after viewing extravagantly priced toys at FAO Schwarz, which they desire but could never afford. The following quote demonstrates the successful completion of the day's lesson, including Sylvia's inner turmoil at being disturbed by inequality, despite a desire to remain indifferent.

“Imagine for a minute what kind of society it is in which some people can spend on a toy what it would cost to feed a family of six or seven. What do you think?”
“I think,” say Sugar, pushing me off her feet like she never done before, cause I whip her ass in a minute, “that this is not much of a democracy if you ask me. Equal chance to pursue happiness means an equal crack at the dough, don’t it?”
Miss Moore is besides herself and I am disgusted with Sugar’s treachery. So I stand on her foot one more time to see if she’ll shove me. She shuts up, and Miss Moore looks at me, sorrowfully I’m thinkin. And somethin weird is goin on, I can feel it in my chest.

Sylvia ultimately declares that "ain’t nobody gonna beat me at nuthin," revealing newly acquired motivation and introspection as the story concludes.