What is the woman doing while she is hanging clothes in Part 2, Chapters 3-4 of 1984?
The woman is singing while she is hanging out her clothes.
Winston is completely confused by the woman's behavior. He cannot understand why she is singing, because the song itself is meaningless, composed by the Party through the use of machines "without any human intervention whatever on an instrument known as a versificator". The words of the song are also meaningless, but the woman sings "so tunefully as to turn the dreadful rubbish into an almost pleasant sound". She seems happy as she does her laundry; it appears "that she would have been perfectly content if the June evening had been endless and the supply of clothes inexhaustible, to remain there for a thousand years, pegging out diapers and singing rubbish", and Winston cannot understand why.
Winston realizes that he "has never heard a member of the Party singing alone and spontaneously". He has been so brainwashed and his life has been so controlled that he cannot conceive of a person singing with happiness just because they can. The Party has regimented life to the point that the people are almost automatons. They are completely detached from ordinary emotions, and so there is nothing in Winston's experience that will allow him to understand how the prole woman can be content with just living, and express her happiness by singing a seemingly meaningless song while performing what appears to him to be the drudgery of household chores (Part 2, Chapter 4).