An argumentative essay and a research paper are generally two different types of written works. It seems that your assignment is for an argumentative essay. A thesis statement that lends itself well to debate takes a strong position on a topic. A research paper also will have a thesis, but the paper itself will be supported with evidence from a wide variety of sources rather than just from within the primary text.
The central character in Tillie Olsen’s story is a mother who has tried to do the right things to raise her daughter, Emily, who is now a young woman about to finish school. The mother reflects on many events in Emily’s life that have brought them to their current relationship. Emily seems to be an indifferent student and may be in trouble at school, as suggested by the letter that a school staffer has sent to her mother.
The mother, who is the first-person narrator, is reflecting on the past more than planning for the future. She seems to regard the future as Emily’s province to determine on her own. Although it becomes clear to the reader that the mother will not actually write a response and send it to the school, the array of topics she covers in her meditations concern diverse aspects of Emily’s upbringing.
In terms of a thesis that a writer could support, it will be necessary to decide what you think the author wants the reader to take away from the story. The mother acknowledges errors that she made in her parenting, but she also provides the context—the United States during the Great Depression—for the lack of services to help single mothers and poor families at that time. One thesis would see the family as deeply troubled and hold the mother responsible for the failures. A different thesis would assess the underlying responsibility for these problems on the society, which provided few or, in some cases, no alternatives for child care and health care that a working mother could access. Both potential theses could be well supported from the evidence in the story.