I am not a professional analyzer, but here is my analysis paper. I hope that it can help others understand the story better.

“Nevertheless,” said the mother calmly, though growing more pale, “this badge hath taught me, -- it daily teaches me,-- it is teaching me at this moment,-- lessons whereof my child may be the wiser and better, albeit they can profit nothing to myself (pg. 98).”
This part of the story is when Hester and Pearl go to visit Governor Bellingham at his mansion, to bring him a pair of gloves, fit to his order. After Hester’s banishment, she began doing seamstress work for many people of the Puritan colony. Mr. Wilson, Roger Chillingsworth, and Reverend Arthur Dimmesdale accompanied Bellingham at his home when Hester came to visit.
While I read this passage for the first time, I had to reread it, and the second time it was better. I imagined that I was Hester, explaining to these men that I deserved to keep my daughter, because her ‘scarlet letter’ has taught her better. While she’s basically begging to keep her daughter, can you imagine how she looked? Flushed and pale as the passage reads, her hands were probably clammy, and her voice probably a little shaky too. I can’t even begin to imagine what it would feel like to walk through town, with, pretty much, a sign exclaiming your sin to everyone. I can’t stand it when people stop and stare, I’d probably make faces at them. During this time women were still subservient, I think that’s the right word, and generally, had no say in what happened to them. These magistrates could easily enough take Pearl away from Esther.
She explains to them that her red letter teaches her always, and it in turn helps her teach her daughter better. Even though teaching her daughter better will not profit her in any way. Do you recall, back at Hester’s cottage one day, when Pearl was throwing rocks at her Mothers ‘scarlet letter, how saddened Hester must have felt. She knows what she’s done was wrong, but I think she knew that when she had to stand before everyone holding Pearl. How has Hester’s experience taught Pearl? Walking through town she’s learned to ignore the snickers, well, almost, because she does throw rocks at them. The older she gets the more she’ll be able to learn and understand from her Mothers experiences. Even though, throughout the book, many people seem to believe that Pearl is just as much as a sinner as her mother. Will Pearl get a fighting chance to prove, or disprove, that her mother’s sins, are not hers as well. Everyone person makes their own decisions, and those choices will directly affect the individual. Our choices, although, do affect others, more particularly our family and loved ones.
I believe that at this point in the story, this passage is a breakthrough for Hester. She’s been living with her sin, but she hasn’t had to answer for it until then. I don’t think she thought they’d take Pearl from her, unless for obvious reasons, but she’s a better mother than that. If Pearl doesn’t learn from her Mothers mistakes, will she also fall into sin? What would happen to Hester, if Pearl were to fall into this sin? Unfortunately, they’d probably hang her. Too often in this time period were people to quick to judge, in my opinion. I don’t think that she should have had to stand in front of everyone, go to jail, or be banished. If this were to happen today, which adultery does, they wouldn’t face the publicized embarrassment that Hester did.